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ELECTROMAGNETIC THEORY 



INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS 

LEONARD I. SCHIFF, CONSULTING EDITO R 

Adler, Bazin, and Schiffer Introduction to General Relativity 

Allis and Herlin Thermodynamics and Statistical Mechanics 

Becker Introduction to Theoretical Mechanics 

Bjorken and Drell Relativistic Quantum Mechanics 

Bjorkin and Drell Relativistic Quantum Fields 

Chodorow and Susskind Fundamentals of Microwave Electronics 

Clark Applied X-rays 

Collin Field Theory of Guided Waves 

Evans The Atomic Nucleus 

Feynman and Hibbs Quantum Mechanics and Path Integrals 

Ginzton Microwave Measurements 

Green Nuclear Physics 

Gurney Introduction to Statistical Mechanics 

Hall Introduction to Electron Microscopy 

Hardy and Perrin The Principles of Optics 

Harnwell Electricity and Electromagnetism 

Harnwell and Livingood Experimental Atomic Physics 

Henley and Thirring Elementary Quantum Field Theory 

Houston Principles of Mathematical Physics 

Kennard Kinetic Theory of Gases 

Lane Superfluid Physics 

Leighton Principles of Modern Physics 

Lindsay Mechanical Radiation 

Livingston and Blewett Particle Accelerators 

Middleton An Introduction to Statistical Communication Theory 

Morse Vibration and Sound 

Morse and Feshbach Methods of Theoretical Physics 

Muskat Physical Principles of Oil Production 

Present Kinetic Theory of Gases 

Read Dislocations in Crystals 

Richtmyer, Kennard, and Lauritsen Introduction to Modern Physics 

Schiff Quantum Mechanics 

Seitz The Modern Theory of Solids 

Slater Introduction to Chemical Physics 

Slater Quantum Theory of Atomic Structure, Vol. I 

Slater Quantum Theory of Atomic Structure, Vol. II 

Slater Quantum Theory of Matter 

Slater Quantum Theory of Molecules and Solids, Vol. 1 

Slater Quantum Theory of Molecules and Solids, Vol. 2 

Slater and Frank Electromagnetism 

Slater and Frank Introduction to Theoretical Physics 

Slater and Frank Mechanics 

Smythe Static and Dynamic Electricity 

Stratton Electromagnetic Theory 

Tinkham Group Theory and Quantum Mechanics 

Townes and Schawlow Microwave Spectroscopy 

White Introduction to Atomic Spectra 

The late F. K. Richtmyer was Consulting Editor of the series from its inception in 
1929 to his death in 1939. Lee A. DuBridge was Consulting Editor from 1939 to 
1946; and G. P. Harnwell from 1947 to 1954. 



ELECTROMAGNETIC 
THEORY 



BY 

JULIUS ADAMS STRATTON 

Professor of Physics 
Massachusetts Institute of Technology 



McGRAW-HILL BOOK COMPANY 

NEW YOBK AND LONDON 
1941 



ELECTROMAGNETIC THEORY 

COPYRIGHT, 1941, BY THE 
McGRAw-HiLL BOOK COMPANY, INC. 



PBINTED IN THE UNITED STATES OP AMERICA 

All rights reserved. This book, or 

parts thereof, may not be reproduced 

in any form without permission of 

the publishers. 



PREFACE 

The pattern set nearly 70 years ago by Maxwell's Treatise on Electric- 
ity and Magnetism has had a dominant influence on almost every subse- 
quent English and American text, persisting to the present day. The 
Treatise was undertaken with the intention of presenting a connected 
account of the entire known body of electric and magnetic phenomena 
from the single point of view of Faraday. Thus it contained little or 
no mention of the hypotheses put forward on the Continent in earlier 
years by Riemann, Weber, Kirchhoff, Helmholtz, and others. It is 
by no means clear that the complete abandonment of these older theories 
was fortunate for the later development of physics. So far as the 
purpose of the Treatise was to disseminate the ideas of Faraday, it was 
undoubtedly fulfilled; as an exposition of the author's own contributions, 
it proved less successful. By and large, the theories and doctrines 
peculiar to Maxwell the concept of displacement current, the identity 
of light and electromagnetic vibrations appeared there in scarcely 
greater completeness and perhaps in a less attractive form than in the 
original memoirs. We find that all of the first volume and a large part 
of the second deal with the stationary state. In fact only a dozen pages 
are devoted to the general equations of the electromagnetic field, 18 to 
the propagation of plane waves and the electromagnetic theory of light, 
and a score more to magnetooptics, all out of a total of 1,000. The 
mathematical completeness of potential theory and the practical utility of 
circuit theory have influenced English and American writers in very 
nearly the same proportion since that day. Only the original and 
solitary genius of Heaviside succeeded in breaking away from this course. 

For an exploration of the fundamental content of Maxwell's equations 
one must turn again to the Continent. There the work of Hertz, Poin- 
car6, Lorentz, Abraham, and Sommerfeld, together with their associates 
and successors, has led to a vastly deeper understanding of physical 
phenomena and to industrial developments of tremendous proportions. 

The present volume attempts a more adequate treatment of variable 
electromagnetic fields and the theory of wave propagation. Some atten- 
tion is given to the stationary state, but for the purpose of introducing 
fundamental concepts under simple conditions, and always with a view 
to later application in the general case. The reader must possess a 
general knowledge of electricity and magnetism such as may be acquired 
from an elementary course based on the experimental laws of Coulomb, 



Vi PREFACE 

Amp&re, and Faraday, followed by an intermediate course dealing with 
the more general properties of circuits, with thermionic and electronic 
devices, and with the elements of electromagnetic machinery, termi- 
nating in a formulation of Maxwell's equations. This book takes up 
at that point. The first chapter contains a general statement of the 
equations governing fields and potentials, a review of the theory of units, 
reference material on curvilinear coordinate systems and the elements of 
tensor analysis, concluding with a formulation of the field equations in 
a space-time continuum. The second chapter is also general in char- 
acter, and much of it may be omitted on a first reading. Here one will 
find a discussion of fundamental field properties that may be deduced 
without reference to particular coordinate systems. A dimensional 
analysis of Maxwell's equations leads to basic definitions of the vectors 
E and B, and an investigation of the energy relations results in expres- 
sions for the mechanical force exerted on elements of charge, current, and 
neutral matter. In this way a direct connection is established between 
observable forces and the vectors employed to describe the structure of a 
field. 

In Chaps. Ill and IV stationary fields are treated as particular cases 
of the dynamic field equations. The subject of wave propagation is 
taken up first in Chap. V, which deals with homogeneous plane waves. 
Particular attention is given to the methods of harmonic analysis, and 
the problem of dispersion is considered in some detail. Chapters VI and 
VII treat the propagation of cylindrical and spherical waves in unbounded 
spaces. A necessary amount of auxiliary material on Bessel functions 
and spherical harmonics is provided, and consideration is given to vector 
solutions of the wave equation. The relation of the field to its source, 
the general theory of radiation, and the outlines of the Kirchhoff-Huygens 
diffraction theory are discussed in Chap. VIII. 

Finally, in Chap. IX, we investigate the effect of plane, cylindrical, 
and spherical surfaces on the propagation of electromagnetic fields. 
This chapter illustrates, in fact, the application of the general theory 
established earlier to problems of practical interest. The reader will 
find here the more important laws of physical optics, the basic theory 
governing the propagation of waves along cylindrical conductors, a 
discussion of cavity oscillations, and an outline of the theory of wave 
propagation over the earth's surface. 

It is regrettable that numerical solutions of special examples could 
not be given more frequently and in greater detail. Unfortunately the 
demands on space in a book covering such a broad field made this imprac- 
tical. The primary objective of the book is a sound expositfon of 
electromagnetic theory, and examples have been chosen with a view to 
illustrating its principles. No pretense is made of an exhaustive treat- 



PREFACE vii 

mcnt of antenna design, transmission-line characteristics, or similar 
topics of engineering importance. It is the author's hope that the 
present volume will provide the fundamental background necessary for 
a critical appreciation of original contributions in special fields and satisfy 
the needs of those who are unwilling to accept engineering formulas 
without knowledge of their origin and limitations. 

Each chapter, with the exception of the first two, is followed by a 
set of problems. There is only one satisfactory way to study a theory, 
and that is by application to specific examples. The problems have been 
chosen with this in mind, but they cover also many topics which it was 
necessary to eliminate from the text. This is particularly true of the 
later chapters. Answers or references are provided in most cases. 

This book deals solely with large-scale phenomena. It is a sore 
temptation to extend the discussion to that fruitful field which Frenkel 
terms the "quasi-microscopic state/ 7 and to deal with the many beautiful 
results of the classical electron theory of matter. In the light of con- 
temporary developments, anyone attempting such a program must soon 
be overcome with misgivings. Although many laws of classical electro- 
dynamics apply directly to submicroscopic domains, one has no basis 
of selection. The author is firmly convinced that the transition must be 
made from quantum electrodynamics toward classical theory, rather 
than in the reverse direction. Whatever form the equations of quantum 
electrodynamics ultimately assume, their statistical average over large 
numbers of atoms must lead to Maxwell's equations. 

The m.k.s. system of units has been employed exclusively. There 
is still the feeling among many physicists that this system is being forced 
upon them by a subversive group of engineers. Perhaps it is, although 
it was Maxwell himself who first had the idea. At all events, it is a good 
system, easily learned, and one that avoids endless confusion in practical 
applications. At the moment there appears to be no doubt of its uni- 
versal adoption in the near future. Help for the tories among us who 
hold to the Gaussian system is offered on page 241. 

In contrast to the stand taken on the m.k.s. system, the author 
has no very strong convictions on the matter of rationalized units. 
Rationalized units have been employed because Maxwell's equations are 
taken as the starting point rather than Coulomb's law, and it seems 
reasonable to make the point of departure as simple as possible. As a 
result of this choice all equations dealing with energy or wave propagation 
are free from the factor 47r. Such relations are becoming of far greater 
practical importance than those expressing the potentials and field 
vectors in terms of their sources. 

The use of the time factor e~ iut instead of e +<w * is another point of 
mild controversy. This has been done because the time factor is invar- 



viii PREFACE 

iably discarded, and it is somewhat more convenient to retain the positive 
exponent e + * R for a positive traveling wave. To reconcile any formula 
with its engineering counterpart, one need only replace i by +j. 

The author has drawn upon many sources for his material and is 
indebted to his colleagues in both the departments of physics and of 
electrical engineering at the Massachusetts Institute of Technology. 
Thanks are expressed particularly to Professor M. F. Gardner whose 
advice on the practical aspects of Laplace transform theory proved 
invaluable, and to Dr. S. Silver who read with great care a part of the 
manuscript. In conclusion the author takes this occasion to express his 
sincere gratitude to Catherine N. Stratton for her constant encourage- 
ment during the preparation of the manuscript and untiring aid in the 
revision of proof. 

JULIUS ADAMS STRATTON. 
CAMBRIDGE, MASS., 
January, 1941. 



CONTENTS 

PAQJD 

PREFACE v 

CHAPTER I 
THE FIELD EQUATIONS 

MAXWELL'S EQUATIONS 1 

1.1 The Field Vectors 1 

1.2 Charge and Current 2 

1.3 Divergence of the Field Vectors 6 

1.4 Integral Form of the Field Equations 6 

MACROSCOPIC PROPERTIES OF MATTER 10 

1.5 The Inductive Capacities e and /x 10 

1.6 Electric and Magnetic Polarization 11 

1.7 Conducting Media 13 

UNITS AND DIMENSIONS 16 

1.8 M.K.S. or Giorgi System 16 

THE ELECTROMAGNETIC POTENTIALS 23 

1.9 Vector and Scalar Potentials 23 

1.10 Conducting Media 26 

1.11 Hertz Vectors, or Polarization Potentials 28 

1.12 Complex Field Vectors and Potentials 32 

BOUNDARY CONDITIONS 34 

1.13 Discontinuities in the Field Vectors 34 

COORDINATE SYSTEMS 38 

1.14 Unitary and Reciprocal Vectors 38 

1.15 Differential Operators 44 

1.16 Orthogonal Systems 47 

1.17 Field Equations in General Orthogonal Coordinates 50 

1.18 Properties of Some Elementary Systems 51 

THE FIELD TENSORS 59 

1.19 Orthogonal Transformations and Their Invariants 59 

1.20 Elements of Tensor Analysis 64 

1.21 Space-time Symmetry of the Field Equations 69 

1.22 The Lorentz Transformation 74 

1.23 Transformation of the Field Vectors to Moving Systems 78 

ix 



X CONTENTS 

PAGB 
CHAPTER II 

STRESS AND ENERGY 

STRESS AND STRAIN IN ELASTIC MEDIA 83 

2.1 Elastic Stress Tensor 83 

2.2 Analysis of Strain 87 

2.3 Elastic Energy and the Relations of Stress to Strain 93 

ELECTROMAGNETIC FORCES ON CHARGES AND CURRENTS 96 

2.4 Definition of the Vectors E and B 96 

2.5 Electromagnetic Stress Tensor in Free Space 97 

2.6 Electromagnetic Momentum 103 

ELECTROSTATIC ENERGY 104 

2.7 Electrostatic Energy as a Function of Charge Density 104 

2.8 Electrostatic Energy as a Function of Field Intensity 107 

2.9 A Theorem on Vector Fields Ill 

2.10 Energy of a Dielectric Body in an Electrostatic Field 112 

2.11 Thomson's Theorem 114 

2.12 Earnshaw's Theorem 116 

2.13 Theorem on the Energy of Uncharged Conductors 117 

MAGNETOSTATTC ENERGY 118 

2.14 Magnetic Energy of Stationary Currents 118 

2.15 Magnetic Energy as a Function of Field Intensity 123 

2.16 Ferromagnetic Materials 125 

2.17 Energy of a Magnetic Body in a Magnetostatic Field 126 

2.18 Potential Energy of a Permanent Magnet 129 

ENERGY FLOW 131 

2.19 Poynting's Theorem 131 

2.20 Complex Poynting Vector 135 

FORCES ON A DIELECTRIC IN AN ELECTROSTATIC FIELD 137 

2.21 Body Forces in Fluids 137 

2.22 Body Forces in Solids 140 

2.23 The Stress Tensor 146 

2.24 Surfaces of Discontinuity 147 

2.25 Electrostriction 149 

2.26 Force on a Body Immersed in a Fluid 151 

FORCES IN THE MAGNETOSTATIC FIELD 153 

2.27 Nonferromagnetic Materials 153 

2.28 Ferromagnetic Materials 155 

FORCES IN THE ELECTROMAGNETIC FIELD 156 

2.29 Force on a Body Immersed in a Fluid 156 

CHAPTER III 

THE ELECTROSTATIC FIELD 

GENERAL PROPERTIES OP AN ELECTROSTATIC FIELD 160 

3.1 Equations of Field and Potential 160 

3.2 Boundary Conditions 163 



CONTENTS 3d 

PAQB 

CALCULATION OF THE FIELD FROM THE CHARGE DISTRIBUTION 165 

3.3 Green's Theorem 165 

3.4 Integration of Poisson's Equation , 166 

3.5 Behavior at Infinity 167 

3.6 Coulomb Field 169 

3.V Convergence of Integrals 170 

EXPANSION OF THE POTENTIAL IN SPHERICAL HARMONICS 172 

3.8 Axial Distributions of Charge 172 

3.9 The Dipole 175 

3.10 Axial Multipoles 176 

3.11 Arbitrary Distributions of Charge 178 

3.12 General Theory of Multipoles 179 

DIELECTRIC POLARIZATION 183 

3.13 Interpretation of the Vectors P and n 183 

DISCONTINUITIES OF INTEGRALS OCCURRING IN POTENTIAL THEORY 185 

3.14 Volume Distributions of Charge and Dipole Moment 185 

3.15 Single-layer Charge Distributions 187 

3.16 Double-layer Distributions 188 

3.17 Interpretation of Green's Theorem 192 

3.18 Images 193 

BOUNDARY-VALUE PROBLEMS 194 

3.19 Formulation of Electrostatic Problems 194 

3.20 Uniqueness of Solution 196 

3.21 Solution of Laplace's Equation 197 

PROBLEM OF THE SPHERE 201 

3.22 Conducting Sphere in Field of a Point Charge 201 

3.23 Dielectric Sphere in Field of a Point Charge 204 

3.24 Sphere in a Parallel Field 205 

PROBLEM OF THE ELLIPSOID 207 

3.25 Free Charge on a Conducting Ellipsoid 207 

3.26 Conducting Ellipsoid in a Parallel Field 209 

3.27 Dielectric Ellipsoid in a Parallel Field 211 

3.28 Cavity Definitions of E and D 213 

3.29 Torque Exerted on an Ellipsoid 215 

PROBLEMS 217 

CHAPTER IV 

THE MAGNETOSTATIC FIELD 
GENERAL PROPERTIES OF A MAGNETOSTATIC FIELD 225 

4.1 Field Equations and the Vector Potential 225 

4.2 Scalar Potential 226 

4.3 Poisson's Analysis 228 

CALCULATION OF THE FIELD OF A CURRENT DISTRIBUTION 230 

4.4 Biot-Savart Law 230 

4.5 Expansion of the Vector Potential ... 233 



Xii CONTENTS 

PAGB 

4.6 The Magnetic Dipole 236 

4.7 Magnetic Shells 237 

A DIGRESSION ON UNITS AND DIMENSIONS 238 

4.8 Fundamental Systems 238 

4.9 Coulomb's Law for Magnetic Matter 241 

MAGNETIC POLARIZATION 242 

4.10 Equivalent Current Distributions 242 

4.11 Field of Magnetized Rods and Spheres 243 

DISCONTINUITIES OF THE VECTORS A AND B 245 

4.12 Surface Distributions of Current 245 

4.13 Surface Distributions of Magnetic Moment 247 

INTEGRATION OF THE EQUATION V X V X A = ju J 250 

4.14 Vector Analogue of Green's Theorem 250 

4.15 Application to the Vector Potential 250 

BOUNDARY-VALUE PROBLEMS 254 

4.16 Formulation of the Magnetostatic Problem 254 

4.17 Uniqueness of Solution 256 

PROBLEM OF THE ELLIPSOID 257 

4.18 Field of a Uniformly Magnetized Ellipsoid 257 

4.19 Magnetic Ellipsoid in a Parallel Field 258 

CYLINDER IN A PARALLEL FIELD 258 

4.20 Calculation of the Field 258 

4.21 Force Exerted on the Cylinder 261 

PROBLEMS 262 

CHAPTER V 

PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA 
PROPAGATION OF PLANE WAVES 268 

5.1 Equations of a One-dimensional Field 268 

5.2 Plane Waves Harmonic in Time 273 

5.3 Plane Waves Harmonic in Space 278 

5.4 Polarization 279 

5.6 Energy Flow 281 

5.6 Impedance 282 

GENERAL SOLUTIONS OF THE ONE-DIMENSIONAL WAVE EQUATION 284 

5.7 Elements of Fourier Analysis 285 

5.8 General Solution of the One-dimensional Wave Equation in a Nondissipa- 
tive Medium 292 

5.9 Dissipative Medium; Prescribed Distribution in Time 297 

5.10 Dissipative Medium; Prescribed Distribution in Space 301 

5.11 Discussion of a Numerical Example 304 

5.12 Elementary Theory of the Laplace Transformation 309 

5.13 Application of the Laplace Transformation to Maxwell's Equations . . . 318 



CONTENTS xiii 

PAOS 

DISPERSION , 321 

5.14 Dispersion in Dielectrics 321 

5.15 Dispersion in Metals 325 

5.16 Propagation in an Ionized Atmosphere 327 

VELOCITIES OP PROPAGATION 330 

5.17 Group Velocity 330 

5.18 Wave-front and Signal Velocities 333 

PROBLEMS 340 

CHAPTER VI 

CYLINDRICAL WAVES 

EQUATIONS OF A CYLINDRICAL FIELD 349 

6.1 Representation by Hertz Vectors 349 

6.2 Scalar and Vector Potentials 351 

6.3 Impedances of Harmonic Cylindrical Fields 354 

WAVE FUNCTIONS OF THE CIRCULAR CYLINDER 355 

6.4 Elementary Waves 355 

6.5 Properties of the Functions Z p (p) 357 

6.6 The Field of Circularly Cylindrical Wave Functions 360 

INTEGRAL REPRESENTATIONS OF WAVE FUNCTIONS 361 

6.7 Construction from Plane Wave Solutions 361 

6.8 Integral Representations of the Functions Z n (p) 364 

6.9 Fourier-Bessel Integrals 369 

6.10 Representation of a Plane Wave 371 

6.11 The Addition Theorem for Circularly Cylindrical Waves 372 

WAVE FUNCTIONS OF THE ELLIPTIC CYLINDER 375 

6.12 Elementary Waves 375 

6.13 Integral Representations 380 

6.14 Expansion of Plane and Circular Waves 384 

PROBLEMS 387 

CHAPTER VII 

SPHERICAL WAVES 

THE VECTOR WAVE EQUATION 392 

7.1 A Fundamental Set of Solutions 392 

7.2 Application to Cylindrical Coordinates . 395 

THE SCALAR WAVE EQUATION IN SPHERICAL COORDINATES 399 

7.3 Elementary Spherical Waves 399 

7.4 Properties of the Radial Functions 404 

7.5 Addition Theorem for the Legendre Polynomials . 406 

7.6 Expansion of Plane Waves 408 

7.7 Integral Representations 409 

7.8 A Fourier-Bessel Integral 411 

7.9 Expansion of a Cylindrical Wave Function 412 

7.10 Addition Theorem for z Q (kR) 413 



xiv CONTENTS 

PAQB 
THE VECTOR WAVE EQUATION IN SPHERICAL COORDINATES 414 

7.11 Spherical Vector Wave Functions 414 

7.12 Integral Representations 416 

7.13 Orthogonality 417 

7.14 Expansion of a Vector Plane Wave 418 

PROBLEMS 420 

CHAPTER VIII 

RADIATION 

THE INHOMOQENEOUS SCALAR WAVE EQUATION 424 

8.1 Kirchhoff Method of Integration 424 

8.2 Retarded Potentials 428 

8.3 Retarded Hertz Vector 430 

A MULTIPOLE EXPANSION 431 

8.4 Definition of the Moments 431 

8.5 Electric Dipole 434 

8.6 Magnetic Dipole 437 

RADIATION THEORY OF LINEAR ANTENNA SYSTEMS 438 

8.7 Radiation Field of a Single Linear Oscillator 438 

8.8 Radiation Due to Traveling Waves 445 

8.9 Suppression of Alternate Phases 446 

8.10 Directional Arrays 448 

8.11 Exact Calculation of the Field of a Linear Oscillator 454 

8.12 Radiation Resistance by the E.M.F. Method 457 

THE KlRCHHOFF-HUYGENS PRINCIPLE 460 

8.13 Scalar Wave Functions 460 

8.14 Direct Integration of the Field Equations 464 

8.15 Discontinuous Surface Distributions 468 

FOUR-DIMENSIONAL FORMULATION OF THE RADIATION PROBLEM 470 

8.16 Integration of the Wave Equation 470 

8.17 Field of a Moving Point Charge 473 

PROBLEMS 477 

CHAPTER IX 

BOUNDARY- VALUE PROBLEMS 

GENERAL THEOREMS 483 

9.1 Boundary Conditions 483 

9.2 Uniqueness of Solution 486 

9.3 Electrodynamic Similitude 488 

REFLECTION AND REFRACTION AT A PLANE SURFACE 490 

9.4 Snell's Laws 490 

9.5 Fresnel's Equations 492 

9.6 Dielectric Media 494 

9.7 Total Reflection 497 

9.8/ Refraction in a Conducting Medium 500 

9.^ Reflection at a Conducting Surface 505 



CONTENTS XV 

PA on 
PLANEpHEBTS 5H 

8/JD Reflection and Transmission Coefficients 511 

_J$\\. Application to Dielectric Media 513 

9.12 Absorbing Layers 515 

SURFACE WAVES 516 

9.13 Complex Angles of Incidence 516 

9.14 Skin Effect 520 

PROPAGATION ALONG A CIRCULAR CYLINDER 524 

9.15 Natural Modes 524 

9.16 Conductor Embedded in a Dielectric 527 

9.17 Further Discussion of the Principal Wave 531 

9.18 Waves in Hollow Pipes 537 

COAXIAL LINES 545 

9.19 Propagation Constant 545 

9.20 Infinite Conductivity 543 

9.21 Finite Conductivity 551 

OSCILLATIONS OP A SPHERE 554 

9.22 Natural Modes 554 

9.23 Oscillations of a Conducting Sphere 558 

9.24 Oscillations in a Spherical Cavity 560 

DIFFRACTION OF A PLANE WAVE BY A SPHERE 563 

9.25 Expansion of the Diffracted Field 563 

9.26 Total Radiation \\ ' 568 

9.27 Limiting Cases 570 

EFFECT OF THE EARTH ON THE PROPAGATION OF RADIO WAVES 573 

9.28 Sommerfeld Solution 573 

9.29 Weyl Solution 577 

9.30 van der Pol Solution 582 

9.31 Approximation of the Integrals 583 

PROBLEMS 588 

APPENDIX I 

A. NUMERICAL VALUES OF FUNDAMENTAL CONST \NTS 601 

B. DIMENSIONS OF ELECTROMAGNETIC QUANTITIES 601 

C. CONVERSION TABLES 602 

APPENDIX II 
FORMULAS FROM VECTOR ANALYSIS 604 

APPENDIX III 

CONDUCTIVITY OF VARIOUS MATERIALS 605 

SPECIFIC INDUCTIVE CAPACITY OF DIELECTRICS 606 

APPENDIX IV 
ASSOCIATED LEGENDRE FUNCTIONS 608 



INDEX. 



609 



ELECTROMAGNETIC THEORY 



CHAPTER I 
THE FIELD EQUATIONS 

A vast wealth of experimental evidence accumulated over the past 
century leads one to believe that large-scale electromagnetic phenomena 
are governed by Maxwell's equations. Coulomb's determination of the 
law of force between charges, the researches of Amp&re on the interaction 
of current elements-, and the observations of Faraday on variable fields 
can be woven into a plausible argument to support this view. The 
historical approach is recommended to the beginner, for it is the simplest 
and will afford him the most immediate satisfaction. In the present 
volume, however, we shall suppose the reader to have completed such a 
preliminary survey and shall credit him with a general knowledge of the 
experimental facts and their theoretical interpretation. Electromagnetic 
theory, according to the standpoint adopted in this book, is the theory of 
Maxwell's equations. Consequently, we shall postulate these equations 
at the outset and proceed to deduce the structure and properties of the 
field together with its relation to the source. No single experiment 
constitutes proof of a theory. The true test of our initial assumptions 
will appear in the persistent, uniform correspondence of deduction with 
observation. 

In this first chapter we shall be occupied with the rather dry business 
of formulating equations and preparing the way for our investigation. 

MAXWELL'S EQUATIONS 

1.1. The Field Vectors. By an electromagnetic field let us under- 
stand the domain of the four vectors E and B, D and H. These vectors 
are assumed to be finite throughout the entire field, and at all ordinary 
points to be continuous functions of position and time, with continuous 
derivatives. Discontinuities in the field vectors or their derivatives 
may occur, however, on surfaces which mark an abrupt change in the 
physical properties of the medium. According to the traditional usage, 
E and H are known as the intensities respectively of the electric and 
magnetic field, D is called the electric displacement and B, the magnetic 
induction. Eventually the field vectors must be defined in terms of the 
experiments by which they can be measured. Until these experiments 

1 



2 THE FIELD EQUATIONS [CHAP. I 

are formulated, there is no reason to consider one vector more funda- 
mental than another, and we shall apply the word intensity to mean 
indiscriminately the strength or magnitude of any of the four vectors 
at a point in space and time. 

The source of an electromagnetic field is a distribution of electric 
charge and current. Since we are concerned only with its macroscopic 
effects, it may be assumed that this distribution is continuous rather 
than discrete, and specified as a function of space and time by the den- 
sity of charge p, and by the vector current density J. 

We shall now postulate that at every ordinary point in space the field 
vectors are subject to the Maxwell equations: 

(1) VXE+f = 0, 

(2) VXH-f = J. 

By an ordinary point we shall mean one in whose neighborhood the 
physical properties of the medium are continuous. It has been noted that 
the transition of the field vectors and their derivatives across a surface 
bounding a material body may be discontinuous; such surfaces must, 
therefore, be excluded until the nature of these discontinuities can be 
investigated. 

1.2. Charge and Current. Although the corpuscular nature of elec- 
tricity is well established, the size of the elementary quantum of charge 
is too minute to be taken into account as a distinct entity in a strictly 
macroscopic theory. Obviously the frontier that marks off the domain 
of large-scale phenomena from those which are microscopic is an arbi- 
trary one. To be sure, a macroscopic element of volume must contain 
an enormous number of atoms ; but that condition alone is an insufficient 
criterion, for many crystals, including the metals, exhibit frequently a 
microscopic "grain" or "mosaic" structure which will be excluded from 
our investigation. We are probably well on the safe side in imposing 
a limit of one-tenth of a millimeter as the smallest admissible element 
of length. There are many experiments, such as the scattering of light 
by particles no larger than 10~ 3 mm. in diameter, which indicate that 
the macroscopic theory may be pushed well beyond the limit suggested. 
Nonetheless, we are encroaching here on the proper domain of quantum 
theory, and it is the quantum theory which must eventually determine 
the validity of our assumptions in microscopic regions. 

Let us suppose that the charge contained within a volume element A0 
is Ag. The charge density at any point within At; will be defined by the 
relation 

(3) A? = P Av. 



SBC. 1.2] CHARGE AND CURRENT 3 

Thus by the charge density at a point we mean the average charge per 
unit volume in the neighborhood of that point. In a strict sense (3) 
does not define a continuous function of position, for Av cannot approach 
zero without limit. Nonetheless we shall assume that p can be repre- 
sented by a function of the coordinates and the time which at ordinary 
points is continuous and has continuous derivatives. The value of the 
total charge obtained by integrating that function over a large-scale 
volume will then differ from the true charge contained therein by a 
microscopic quantity at most. 

Any ordered motion of charge constitutes a current. A current dis- 
tribution is characterized by a vector field which specifies at each point 
not only the intensity of the flow but also its direction. As in the study 
of fluid motion, it is convenient to imagine streamlines traced through 
the distribution and everywhere tangent to the direction of flow. Con- 
sider a surface which is orthogonal to a system of streamlines. The 
current density at any- point on this surface is then defined as a vector J 
directed along the streamline through the point and equal in magnitude 
to the charge which in unit time crosses unit area of the surface in the 
vicinity of the point. On the other hand the current / across any surface 
S is equal to the rate at which charge crosses that surface. If n is the 
positive unit normal to an element Aa of $, we have 

(4) AI = J n Aa. 

Since Aa is a macroscopic element of area, Eq. (4) does not define the 
current density with mathematical rigor as a continuous function of 
position, but again one may represent the distribution by such a function 
without incurring an appreciable error. The total current through S is, 
therefore, 

jm 

n da. 

Since electrical charge may be either positive or negative, a convention 
must be adopted as to what constitutes a positive current. If the flow 
through an element of area consists of positive charges whose velocity 
vectors form an angle of less than 90 deg. with the positive normal n, 
the current is said to be positive. If the angle is greater than 90 deg., the 
current is negative. Likewise if the angle is less than 90 deg. but the 
charges are negative, the current through the element is negative. In 
the case of metallic conductors the carriers of electricity are presumably 
negative electrons, and the direction of the current density vector is 
therefore opposed to the direction of electron motion. 

Let us suppose now that the surface S of Eq. (5) is closed. We shall 
adhere to the customary convention that the positive normal to a closed 



4 THE FIELD EQUATIONS [CHAP. I 

surface is drawn outward. In virtue of the definition of current as the 
flow of charge across a surface, it follows that the surface integral of the 
normal component of J over S must measure the loss of charge from the 
region within. There is no experimental evidence to indicate that under 
ordinary conditions charge may be either created or destroyed in macro- 
scopic amounts. One may therefore write 

(6) 

where V is the volume enclosed by $, as a relation expressing the con- 
servation of charge. The flow of charge across the surface can originate 
in two ways. The surface S may be fixed in space and the density p 
be some function of the time as well as of the coordinates; or the charge 
density may be invariable with time, while the surface moves in some 
prescribed manner. In this latter event the right-hand integral of (6) 
is a function of time in virtue of variable limits. If, however, the surface 
is fixed and the integral convergent, one may replace d/dt by a partial 
derivative under the sign of integration. 



(7) 



f J.nda=~ f % 
Js Jvdt 



We shall have frequent occasion to make use of the divergence theorem 
of vector analysis. Let A(x, t/, 2) be any vector function of position 
which together with its first derivatives is continuous throughout a 
volume V and over the bounding surface S. The surface S is regular 
but otherwise arbitrary. 1 Then it can be shown that 

(8) f s A n da = j^ V A dv. 

As a matter of fact, this relation may be advantageously used as a 
definition of the divergence. To obtain the value of V A at a point P 
within F, we allow the surface S to shrink about P. When the volume V 
has become sufficiently small, the integral on the right may be replaced 
by W A, and we obtain 

(9) V A = Km ^7 I A n da. 

s-+o V Js 

1 A regular element of arc is represented in parametric form by the equations 
X = x(t\y *=* y(t), z =* z(t) such that in the interval a ^ t ^ b x, y, z are continuous, 
single-valued functions of t with continuous derivatives of all orders unless otherwise 
restricted. A regular curve is constructed of a finite number of such arcs joined end 
to end but such that the curve does not cross itself. Thus a regular curve has no 
double points and is piecewise differentiable. A regular surface element is a portion 
of surface whose projection on a properly oriented plane is the interior of a regular 
closed curve. Hence it does not intersect itself. Cf. Kellogg, "Foundations of Poten- 
tial Theory," p. 97, Springer, 1929. 



SBC, 1.2] CHARGE AND CURRENT 5 

The divergence of a vector at a point is, therefore, to be interpreted as the 
integral of its normal component over an infinitesimally small surface 
enclosing that point, divided by the enclosed volume. The flux of a 
vector through a closed surface is a measure of the sources within; hence 
rhe divergence determines their strength at a point. Since S has been 
shrunk close about P, the value of A at every point on the surface may 
be expressed analytically in terms of the values of A and its derivatives 
at P, and consequently the integral in (9) may be evaluated, leading in 
the case of rectangular coordinates to 



On applying this theorem to (7) the surface integral is transformed 
to the volume integral 



Now the integrand of (11) is a continuous function of the coordinates 
and hence there must exist small regions within which the integrand does 
not change sign. If the integral is to vanish for arbitrary volumes V, it 
is necessary that the integrand be identically zero. The differential 
equation 

(12) T.J + I-O 

expresses the conservation of charge in the neighborhood of a point. 
By analogy with an equivalent relation in hydrodynamics, (12) is fre- 
quently referred to as the equation of continuity. 

If at every point within a specified region the charge density is con- 
stant, the current passing into the region through the bounding surface 
must at all times equal the current passing outward. Over the bounding 
surface S we have 

(13) 

and at every interior point 

(14) V J = 0. 

Any motion characterized by vector or scalar quantities which are 
independent of the time is said to be steady, or stationary. A steady- 
state flow of electricity is thus defined by a vector J which at every point 
within the region is constant in direction and magnitude. In virtue of 
the divergenceless character of such a current distribution, it follows 



8 THE FIELD EQUATIONS [CHAP. I 

that in the steady state all streamlines, or current filaments, close upon 
themselves. The field of the vector J is solenoidal. 

1.3. Divergence of the Field Vectors. Two further conditions satis- 
fied by the vectors B and D may be deduced directly from Maxwell's 
equations by noting that the divergence of the curl of any vector vanishes 
identically. We take the divergence of Eq. (1) and obtain 

(15) v.f = |v.B-0. 

The commutation of the operators V and d/dt is admissible, for at an 
ordinary point B and all its derivatives are assumed to be continuous. 
It follows from (15) that at every point in the field the divergence of B 
is constant. If ever in its past history the field has vanished, this con- 
stant must be zero and, since one may reasonably suppose that the 
initial generation of the field was at a time not infinitely remote, we 
conclude that 

(16) V B = 0, 

and the field of B is therefore solenoidal. 

Likewise the divergence of Eq. (2) leads to 

(i?) VJ + |V.D = O, 

or, in virtue of (12), to 

(18) I (V D - p) = 0. 

If again we admit that at some time in its past or future history the field 
may vanish, it is necessary that 

(19) V - D = p. 

The charges distributed with a density p constitute the sources of the 
vector D. 

The divergence equations (16) and (19) are frequently included as 
part of Maxwell's system. It must be noted, however, that if one assumes 
the conservation of charge, these are not independent relations. 

1.4. Integral Form of the Field Equations. The properties of an 
electromagnetic field which have been specified by the differential equa- 
tions (1), (2), (16), and (19) may also be expressed by an equivalent 
system of integral relations. To obtain this equivalent system, we apply 
a second fundamental theorem of vector analysis. 

According to Stokes' theorem the line integral of a vector taken 
about a closed contour can be transformed into a surface integral extended 



SBC. 1.4] INTEGRAL FORM OF THE FIELD EQUATIONS 7 

over a surface bounded by the contour. The contour C must either be 
regular or be resolvable into a finite number of regular arcs, and it is 
assumed that the otherwise arbitrary surface S bounded by C is two- 
sided and may be resolved into a finite number of regular elements. The 
positive side of the surface S is related to the positive direction of circu- 
lation on the contour by the usual convention that an observer, moving 
in a positive sense along C, will have the positive side of S on his left. 
Then if A(z, y, z) is any vector function of position, which together with 
its first derivatives is continuous at all points of S and C, it may be shown 
that 

(20) 

where ds is an element of length along C and n is a unit vector normal to 
the positive side of the element of area da. This transformation can 
also be looked upon as an equation defining the curl. To determine the 
value of V X A at a point P on fif, we allow the contour to shrink about P 
until the enclosed area S is reduced to an infinitesimal element of a plane 
whose normal is in the direction specified by n. The integral on the 
right is then equal to ( V X A) n/S, plus infinitesimals of higher order. The 
projection of the vector V X A in the direction of the normal is, therefore, 

(21) (V X A) n = lim ~ f A - ds. 

c-o o Jc 

The curl of a vector at a point is to be interpreted as the line integral of 
that vector about an infinitesimal path on a surface containing the point, 
per unit of enclosed area. Since A has been assumed analytic in the 
neighborhood of P, its value at any point on C may be expressed in 
terms of the values of A and its derivatives at P, so that the evaluation of 
the line integral in (21) about the infinitesimal path can actually be 
carried out. In particular, if the element S is oriented parallel to the 
2/z-coordinate plane, one finds for the x-component of the curl 

(22) (V X A). - ^ - 
Proceeding likewise for the y- and 2-components we obtaip 

(23) vxA = 
v 

i j k 

1 A 1 

dx dy dz 

jflijt IJiJJ A*. 



8 



THE FIELD EQUATIONS 



[CHAP. I 



Let us now integrate the normal component of the vector SB/dt over 
any regular surface S bounded by a closed contour C. From (1) and 
(20) it follows that 



(24) 



f E ds + f 
Jc Js 



n da = 0. 



If the contour is fixed, the operator d/dt may be brought out from under 
the sign of integration. 

(25) I E-ds = ~ I B-ncta, 

Jc ot Js 

By definition, the quantity 



(26) 



-I 



B nda 





FIG. 1. Convention relating 
direction of the positive normal 
n to the direction of circulation 
about a contour C. 



is the magnetic flux, or more specifically the flux of the vector B through 
the surface. According to (25) the line integral of the vector E about any 

closed, regular curve in the field is equal 
to the time rate of decrease of the magnetic 
flux through any surface spanning that 
curve. The relation between the direction 
of circulation about a contour and the posi- 
tive normal to a surface bounded by it is 
illustrated in Fig. 1. A positive direction 
about C is chosen arbitrarily and the flux < 
is then positive or negative according to 
the direction of the lines of B with respect 
to the normal. The time rate of change of <$> is in turn positive or nega- 
tive as the positive flux is increasing or decreasing. 

We recall that the application of Stokes' theorem to Eq. (1) is valid 
only if the vector E and its derivatives are continuous at all points of S 
and C. Since discontinuities in both E and B occur across surfaces 
marking sudden changes in the physical properties of the medium, the 
question may be raised as to what extent (25) represents a general law 
of the electromagnetic field. One might suppose, for example, that the 
contour linked or pierced a closed iron transformer core. To obviate 
this difficulty it may be imagined that at the surface of every material 
body in the field the physical properties vary rapidly but continuously 
within a thin boundary layer from their values just inside to their values 
just outside the surface. In this manner all discontinuities are eliminated 
from the field and (25) may be applied to every closed contour. 

The experiments of Faraday indicated that the relation (25) holds 
whatever the cause of flux variation. The partial derivative implies a 



SBC. 1.4] INTEGRAL FORM OF THE FIELD EQUATIONS 9 

variable flux density threading a fixed contour, but the total flux can 
likewise be changed by a deformation of the contour. To take this into 
account the Faraday law is written generally in the form 



(27) 



I E ds = - -j 4 I B n da. 
Jc at Js 



It can be shown that (27) is in fact a consequence of the differential 
field equations, but the proof must be based on the electrodynamics of 
moving bodies which will be touched upon in Sec. 1.22. 

In like fashion Eq. (2) may be replaced by an equivalent integral 
relation, 

(28) I H ds = I + ^ I D n da. 

Jc at Js 

where I is the total current linking the contour as defined in (5). In the 
steady state, the integral on the right is zero and the conduction current / 
through any regular surface is equal to the line integral of the vector H 
about its contour. If, however, the field is variable, the vector d'D/di 
has associated with it a field H exactly equal to that which would be 
produced by a current distribution of density 



To this quantity Maxwell gave the name "displacement current/' a term 
which we shall occasionally employ without committing ourselves as 
yet to any particular interpretation of the vector D. 

The two remaining field equations (16) and (19) can be expressed in 
an equivalent integral form with the help of the divergence theorem. 
One obtains 



(30) B n da = 0, 

stating that the total flux of the vector B crossing any closed, regular 
surface is zero, and 

(31) 

according to which the flux of the vector D through a closed surface is 
equal to the total charge q contained within. The circle through the 
sign of integration is frequently employed to emphasize the fact that a 
contour or surface is closed. 



10 THE FIELD EQUATIONS [CHAP. I 

MACROSCOPIC PROPERTIES OF MATTER 

1.5. The Inductive Capacities e and y. No other assumptions have 
been made thus far than that an electromagnetic field may be charac- 
terized by four vectors E, B, D, and H, which at ordinary points satisfy 
Maxwell's equations, and that the distribution of current which gives 
rise to this field is such as to ensure the conservation of charge. Between 
the five vectors E, B, D, H, J there are but two independent relations, the 
equations (1) and (2) of the preceding section, and we are therefore obliged 
to impose further conditions if the system is to be made determinate. 

Let us begin with the assumption that at any given point in the field, 
whether in free space or within matter, the vector D may be represented 
as a function of E and the vector H as a function of B. 

(1) D D(E), H = ff(B). 

The nature of these functional relations is to be determined solely by the 
physical properties of the medium in the immediate neighborhood of the 
specified point. Certain simple relations are of most common occurrence. 

1. In free space, D differs from E only by a constant factor, as does H 
from B. Following the traditional usage, we shall write 

(2) D - E, H = - B. 

Mo 

The values and the dimensions of the constants 6 and ju will depend 
upon the system of units adopted. In only one of many wholly arbitrary 
systems does D reduce to E and H to B in empty space. 

2. If the physical properties of a body in the neighborhood of some 
interior point are the same in all directions, the body is said to be iso- 
tropic. At every point in an isotropic medium D is parallel to E and H 
is parallel to B. The relations between the vectors, moreover, are linear 
in almost all the soluble problems of electromagnetic theory. For the 
isotropic, linear case we put then 

(3) D = eE, H = iB. 

M 

The factors c and ju, will be called the inductive capacities of the medium. 
The dimensionless ratios 

(4) K. = 1, K m = , 

*o A*o 

are independent of the choice of units and will be referred to as the 
specific inductive capacities. The properties of a homogeneous medium 
are constant from point to point and in this case it is customary to refer 



SBC. 1.6] ELECTRIC AND MAGNETIC POLARIZATION 11 

to K as the dielectric constant and to K m as the permeability. In general, 
however, one must look upon the inductive capacities as scalar functions 
of position which characterize the electromagnetic properties of matter 
in the large. 

3. The properties of anisotropic matter vary in a different manner 
along different directions about a point. In this case the vectors D and 
E, H and B are parallel only along certain preferred axes. If it may be 
assumed that the relations are still linear, as is usually the case, one 
may express each rectangular component of D as a linear function of the 
three components of E. 



-f- 
(5) Dy 



The coefficients ,& of this linear transformation are the components of a 
symmetric tensor. An analogous relation may be set up between the 
vectors H and B, but the occurrence of such a linear anisotropy in what 
may properly be called macroscopic problems is rare. 

The distinction between the microscopic and macroscopic viewpoints 
is nowhere sharper than in the interpretation of these parameters e and /*, 
or their tensor equivalents. A microscopic theory must deduce the 
physical properties of matter from its atomic structure. It must enable 
one to calculate not only the average field that prevails within a body but 
also its local value in the neighborhood of a specific atom. It must tell 
us how the atom will be deformed under the influence of that local field, 
and how the aggregate effect of these atomic deformations may be 
represented in the large by such parameters as e and ju. 

We, on the other hand, are from the present standpoint sheer behav- 
iorists. Our knowledge of matter is, to use a large word, purely phe- 
nomenological. Each substance is to be characterized electromagnetically 
in terms of a minimum number of parameters. The dependence of the 
parameters e and /x on such physical variables as density, temperature, 
and frequency will be established by experiment. Information given by 
such measurements sheds much light on the internal structure of matter, 
but the internal structure is not our present concern. 

1.6. Electric and Magnetic Polarization. To describe the electro- 
magnetic state of a sample of matter, it will prove convenient to intro- 
duce two additional vectors. We shall define the electric and magnetic 
polarization vectors by the equations 

(6) P = D - e E, M = ~ B - H. 

Mo 

The polarization vectors are thus definitely associated with matter and 



12 THE FIELD EQUATIONS 

vanish in free space. By means of these relations let us now eliminate 
D and H from the field equations. There results the system 



VXE + .0, 



(7) 



V X B - eoMo^ = MO (j + ^ + V X M), 

V B = 0, V E = - (p - V P), 





which we are free to interpret as follows: the presence of rigid material 
bodies in an electromagnetic field may be completely accounted for by an 
equivalent distribution of charge of density V P, and an equivalent 

dP 

distribution of current of density - + V X M. 

ot 

In isotropic media the polarization vectors are parallel to the corre- 
sponding field vectors, and are found experimentally to be proportional 
to them if ferromagnetic materials are excluded. The electric and 
magnetic susceptibilities Xe and \ m are defined by the relations 

(8) P = x^ E, M = x-H. 

Logically the magnetic polarization M should be placed proportional to B. 
Long usage, however, has associated it with H and to avoid confusion 
on a matter which is really of no great importance we adhere to this 
convention. The susceptibilities x and \ m defined by (8) are dimension- 
less ratios whose values are independent of the system of units employed. 
In due course it will be shown that E and B are force vectors and in this 
sense are fundamental. D and H are derived vectors associated with 
the state of matter. The polarization vector P has the dimensions of D, 
not E, while M and H are dimensionally alike. From (3), (6), and (8) it 
follows at once that the susceptibilities are related to the specific induc- 
tive capacities by the equations 

(9) Xe = * 1, Xm = *rn ~ 1. 

In anisotropic media the susceptibilities are represented by the com- 
ponents of a tensor. 

It will be a part of our task in later chapters to formulate experiments 
by means of which the susceptibility of a substance may be accurately 
measured. Such measurements show that the electric susceptibility is 
always positive. In gases it is of the order of 0.0006 (air), but in liquids 
and solids it may attain values as large as 80 (water). An inherent 
difference in the nature of the vectors P and M is indicated by the fact 
that the magnetic susceptibility x may be either positive or negative. 
Substances characterized by a positive susceptibility are said to be 



SEC. 1.7] CONDUCTING MEDIA 13 

paramagnetic, whereas those whose susceptibility is negative are called 
diamagnetic. The metals of the ferromagnetic group, including iron, 
nickel, cobalt, and their alloys, constitute a particular class of substances 
of enormous positive susceptibility, the value of which may be of the 
order of many thousands. In view of the nonlinear relation of M to H 
peculiar to these materials, the susceptibility %m must now be interpreted 
as the slope of a tangent to the M-H curve at a point corresponding to a 
particular value of H. To include such cases the definition of suscepti- 
bility is generalized to 

x = 

Xm 



The susceptibilities of all nonferromagnetic materials, whether para- 
magnetic or diamagnetic, are so small as to be negligible for most practical 
purposes. 

Thus far it has, been assumed that a functional relation exists 
between the vector P or M and the applied field, and for this reason 
they may properly be called the induced polarizations. Under certain 
conditions, however, a magnetic field may be associated with a ferro- 
magnetic body in the absence of any external excitation. The body is 
then said to be in a state of permanent magnetization. We shall main- 
tain our initial assumption that the field both inside and outside the 
magnet is completely defined by the vectors B and H. But now the 
difference of these two vectors at an interior point is a fixed vector M , 
which may be called the intensity of magnetization and which bears no 
functional relationship to H. On the contrary the magnetization M 
must be interpreted as the source of the field. If an external field is 
superposed on the field of a permanent magnet, the intensity of magneti- 
zation will be augmented by the induced polarization M. At any interior 
point we have, therefore, 

(11) B = M o(H + M + Mo). 

Of this induced polarization we can only say for the present that it is a 
function of the resultant H prevailing at the same point. The relation 
of the resultant field within the body to the intensity of an applied field 
generated by external sources depends not only on the magnetization 
M but also upon the shape of the body. There will be occasion to 
examine this matter more carefully in Chap. IV. 

1.7. Conducting Media. To Maxwell's equations there must now 
be added a third and last empirical relation between the current density 
and the field. We shall assume that at any point within a liquid or 
solid the current density is a function of the field E. 

(12) J = J(E). 



14 THE FIELD EQUATIONS [CHAP. I 

The distribution of current in an ionized, gaseous medium may depend 
also on the intensity of the magnetic field, but since electromagnetic 
phenomena in gaseous discharges are in general governed by a multitude 
of factors other than those taken into account in the present theory, we 
shall exclude such cases from further consideration. 1 

Throughout a remarkably wide range of conditions, in both solids 
and weakly ionized solutions, the relation (12) proves to be linear. 

(13) J - <rE. 

The factor cr is called the conductivity of the medium. The distinction 
between good and poor conductors, or insulators, is relative and arbitrary. 
All substances exhibit conductivity to some degree but the range of 
observed values of a is tremendous. The conductivity of copper, for 
example, is some 10 7 times as great as that of such a "good" conductor 
as sea water, and 10 19 times that of ordinary glass. In Appendix III 
will be found an abbreviated table of the conductivities of representative 
materials. 

Equation (13) is simply Ohm's law. Let us imagine, for example, a 
stationary distribution of current throughout the volume of any con- 
ducting medium. In virtue of the divergenceless character of the flow 
this distribution may be represented by closed streamlines. If a and 6 
are two points on a particular streamline and ds is an element of its 
length, we have 



f E-rfs = ( b l 
J* Ja <r 



(14) E ds = l.dn. 

Ja Ja 0- 

A bundle of adjacent streamlines constitutes a current filament or tube. 
Since the flow is solenoidal, the current 7 through every cross section of 
the filament is the same. Let S be the cross-sectional area of the filament 
on a plane drawn normal to the direction of flow. S need not be infini- 
tesimal, but is assumed to be so small that over its area the current 
density is uniform. Then SJ ds = / ds, and 

C b C b i 

(15) I E ds = I I -~ ds. 

Ja Ja <TO 

The factor, 



1 It is true that to a very slight degree the current distribution in a liquid or solid 
conductor may be modified by an impressed magnetic field, but the magnitude of this 
so-called Hall effect is so small that it may be ignored without incurring an appreciable 
error. 



SEC. 1.7] CONDUCTING MEDIA 15 

is equal to the resistance of the filament between the points a and 6. 
The resistance of a linear section of homogeneous conductor of uniform 
cross section S and length I is 

(17) B-i 

a formula which is strictly valid only in the case of stationary currents. 
Within a region of nonvanishing conductivity there can be no permanent 
distribution of free charge. This fundamentally important theorem can 
be easily demonstrated when the medium is homogeneous and such that 
the relations between D and E and J and E are linear. By the equation 
of continuity, 



On the other hand in a. homogeneous medium 

(19) V-E = ip, 
which combined with (18) leads to 

(20) ! + ^ p = 0. 

The density of charge at any instant is, therefore, 

(21) p = p e~' , 

the constant of integration p being equal to the density at the time t = 0. 
The initial charge distribution throughout the conductor decays expo- 
nentially with the time at every point and in a manner wholly inde- 
pendent of the applied field. If the charge density is initially zero, it 
remains zero at all times thereafter. 
The time 

(22) r = 1 

required for the charge at any point to decay to 1/e of its original value 
is called the relaxation time. In all but the poorest conductors r is 
exceedingly small. Thus in sea water the relaxation time is about 
2 X 10~ 10 sec. ; even in such a poor conductor as distilled water it is not 
greater than 10~ 6 sec. In the best insulators, such as fused quartz, it 
may nevertheless assume values exceeding 10 6 sec., an instance of the 
extraordinary range in the possible values of the parameter cr. 



16 THE FIELD EQUATIONS [CHAP. I 

Let us suppose that at t = a charge is concentrated within a small 
spherical region located somewhere in a conducting body. At every 
other point of the conductor the charge density is zero. The charge 
within the sphere now begins to fade away exponentially, but according 
to (21) no charge can reappear anywhere within the conductor. What 
becomes of it? Since the charge is conserved, the decay of charge 
within the spherical surface must be accompanied by an outward flow, 
or current. No charge can accumulate at any other interior point; hence 
the flow must be divergenceless. It will be arrested, however, on the 
outer surface of the conductor and it is here that we shall rediscover the 
charge that has been lost from the central sphere. This surface charge 
makes its appearance at the exact instant that the interior charge begins 
to decay, for the total charge is constant. 

UNITS AND DIMENSIONS 

1.8. The M.K.S. or Giorgi System. An electromagnetic field thus 
far is no more than a complex of vectors subject to a postulated system of 
differential equations. To proceed further we must establish the physical 
dimensions of these vectors and agree on the units in which they are 
to be measured. 

In the customary sense, an " absolute" system of units is one in which 
every quantity may be measured or expressed in terms of the three 
fundamental quantities mass, length, and time. Now in electromagnetic 
theory there is an essential arbitrariness in the matter of dimensions 
which is introduced with the factors CQ and /*o connecting D and E, H 
and B respectively in free space. No experiment has yet been imagined 
by means of which dimensions may be attributed to either c or /z as 
an independent physical entity. On the other hand, it is a direct conse- 
quence of the field equations that the quantity 

(1) c = -^ 



shall have the dimensions of a velocity, and every arbitrary choice of 
and JJLQ is subject to this restriction. The magnitude of this velocity 
cannot be calculated a priori, but by suitable experiment it may be 
measured. The value obtained by the method of Rosa and Dorsey of 
the Bureau of Standards and corrected by Curtis 1 in 1929 is 

(2) c = *-. = 2.99790 X 10 8 meters/sec., 

V o 



1 ROSA and DORSET, A New Determination of the Ratio of the Electrostatic Unit 
of Electricity, Bur. Standards, Bull. 3, p. 433, 1907. CURTIS, Bur. Standards J. 
Research, 3, 63, 1929. 



SEC. 1.8] THE M.K.S. OR GIORGI SYSTEM 17 

or for all practical purposes 

(3) c = 3 X 10 8 meters/sec. 

Throughout the early history of electromagnetic theory the absolute 
electromagnetic system of units was employed for all scientific investiga- 
tions. In this system the centimeter was adopted as the unit of length, 
the gram as the unit of mass, the second as the unit of time, and as a 
fourth unit the factor jz was placed arbitrarily equal to unity and con- 
sidered dimensionless. The dimensions of CQ were then uniquely deter- 
mined by (1) and it could be shown that the units and dimensions of 
every other quantity entering into the theory might be expressed in 
terms of centimeters, grams, seconds, and p, . Unfortunately, this abso- 
lute system failed to meet the needs of practice. The units of resistance 
and of electromotive force were, for example, far too small. To remedy 
this defect a practical system was adopted. Each unit of the practical 
system had the dimensions of the corresponding electromagnetic unit and 
differed from it in magnitude by a power of ten which, in the case of 
voltage and resistance at least, was wholly arbitrary. The practical 
units have the great advantage of convenient size and they are now 
universally employed for technical measurements and computations. 
Since they have been defined as arbitrary multiples of absolute units, they 
do not, however, constitute an absolute system. Now the quantities 
mass, length, and time are fundamental solely because the physicist has 
found it expedient to raise them to that rank. That there are other 
fundamental quantities is obvious from the fact that all electromagnetic 
quantities cannot be expressed in terms of these three alone. The 
restriction of the term " absolute" to systems based on mass, length, and 
time is, therefore, wholly unwarranted; one should ask only that such a 
system be self-consistent and that every quantity be defined in terms of 
a minimum number of basic, independent units. The antipathy of 
physicists in the past to the practical system of electrical units has been 
based not on any firm belief in the sanctity of mass, length, and time, 
but rather on the lack of self-consistency within that system. 

Fortunately a most satisfactory solution has been found for this 
difficulty. In 1901 Giorgi, 1 pursuing an idea originally due to Maxwell, 
called attention to the fact that the practical system could be converted 
into an absolute system by an appropriate choice of fundamental units. 
It is indeed only necessary to choose for the unit of length the inter- 

1 GIORGI: Unita Razionali di Elettromagnetismo, Atti dell' A.E.I., 1901. An 
historical review of the development of the practical system, including a report of the 
action taken at the 1935 meeting of the International Electro technical Commission 
and an extensive bibliography is given by Kennelly, /. Inst. Elec. Engrs., 78, 235- 
245, 1936. See also GLAZEBBOOK, The M.K.S. System of Electrical Units, /. Inst. 
Elec. Engrs., 78, pp. 245-247. 



18 THE FIELD EQUATIONS [CHAP. I 

national meter, for the unit of mass the kilogram, for the unit of time the 
second, and as a fourth unit any electrical quantity belonging to the 
practical system such as the coulomb, the ampere, or the ohm. From 
the field equations it is then possible to deduce the units and dimensions 
of every electromagnetic quantity in terms of these four fundamental 
units. Moreover the derived quantities will be related to each other 
exactly as in the practical system and may, therefore, be expressed in 
practical units. In particular it is found that the parameter /* must 
have the value 47r X 10~ 7 , whence from (1) the value of e may be calcu- 
lated. Inversely one might equally well assume this value of MO as a 
fourth basic unit and then deduce the practical series from the field 
equations. 

At a plenary session in June, 1935, the International Electrotechnical 
Commission adopted unanimously the m.k.s. system of Giorgi. Certain 
questions, however, still remain to be settled. No official agreement 
has as yet been reached as to the fourth fundamental unit. Giorgi him- 
self recommended that the ohm, a material standard defined as the 
resistance of a specified column of mercury under specified conditions 
of pressure and temperature, be introduced as a basic quantity. If 
Mo = 4?r X 10~ 7 be chosen as the fourth unit and assumed dimensionless, 
all derived quantities may be expressed in terms of mass, length, and 
time alone, the dimensions of each being identical with those of the corre- 
sponding quantity in the absolute electromagnetic system and differing 
from them only in the size of the units. This assumption leads, however, 
to fractional exponents in the dimensions of many quantities, a direct 
consequence of our arbitrariness in clinging to mass, length, and time 
as the sole fundamental entities. In the absolute electromagnetic sys- 
tem, for example, the dimensions of charge are grams* centimeters*, an 
irrationality which can hardly be physically significant. These fractional 
exponents are entirely eliminated if we choose as a fourth unit the 
coulomb; for this reason, charge has been advocated at various times as a 
fundamental quantity quite apart from the question of its magnitude. 1 
In the present volume we shall adhere exclusively to the meter-kilogram- 
second-coulomb system. A subsequent choice by the I. E.G. of some 
other electrical quantity as basic will in nowise affect the size of our units 
or the form of the equations. 2 

x See the discussion by WALLOT: Elektrotechnische Zeitschrift, Nos. 44r-46, 1922. 
Also SOMMEBFELD: "Ueber die Electromagnetischen Einheiten," pp. 157-165, Zeeman 
Verhandelingen, Martinus Nijhoff, The Hague, 1935; Physik. Z. 36, 81^820, 1935. 

2 No ruling has been made as yet on the question of rationalization and opinion 
seems equally divided in favor and against. If one bases the theory on Maxwell's 
equations, it seems definitely advantageous to drop the factors 4?r which in unrational- 
ized systems stand before the charge and current densities. A rationalized system 
will be employed in this book. 



SBC. 1.8] THE M.K.S. OR GIORGI SYSTEM 19 

To demonstrate that the proposed units do constitute a self-consistent 
system let us proceed as follows. The unit of current in the m.k.s. 
system is to be the absolute ampere and the unit of resistance is to be 
the absolute ohm. These quantities are to be such that the work 
expended per second by a current of 1 amp. passing through a resistance 
of 1 ohm is 1 joule (absolute). If R is the resistance of a section of 
conductor carrying a constant current of I amp., the work dissipated in 
heat in t sec. is 

(4) W = PRt joules. 

By means of a calorimeter the heat generated may be measured and thus 
one determines the relation of the unit of electrical energy to the unit 
quantity of heat. It is desired that the joule defined by (4) be identical 
with the joule defined as a unit of mechanical work, so that in the electrical 
as well as in the mechanical case 

(5) * 1 joule = 0.2389 gram-calorie (mean). 

Now we shall define the ampere on the basis of the equation of continuity 
(6), page 4, as the current which transports across any surface 1 coulomb 
in 1 sec. Then the ohm is a derived unit whose magnitude and dimensions 
are determined by (4) : 

,-v -.in watt - kilogram meter 2 

(6) 1 ohm = 1 - 2 = 1 - r PS - j* 

ampere 2 coulomb 2 second 

since 1 watt is equal to 1 joule/sec. The resistivity of a medium is 
defined as the resistance measured between two parallel faces of a unit 
cube. The reciprocal of this quantity is the conductivity. The dimen- 
sions of v follow from Eq. (17), page 15. 

/rr v . ., - , , . ., 1 . coulomb 2 second 

(7) 1 unit of conductivity = -r - T = 1 rn - 1 r 
N ' ohm meter kilogram meter 3 

In the United States the reciprocal ohm is usually called the mho, 
although the name Siemens has been adopted officially by the I.E.C. 
The unit of conductivity is therefore 1 siemens/meter. 
The volt will be defined simply as 1 watt/amp., or 

(8) 1 volt = 1 watt = 1 kilogram meter 2 
^ ampere coulomb second 2 

Since the unit of current density is 1 amp. /meter 2 , we deduce from the. 
relation J = <rE that 



/r^ i -j. r T? i wa *t i volt . kilogram meter 

(9) 1 unit of E = 1 - = 1 = 1 - i^r - TV 
ampere meter meter coulomb second 2 



20 THE FIELD EQUATIONS [CHAP. I 

The power expended per unit volume by a current of density J is there- 
fore E J watts/meter 3 . It will be noted furthermore that the product 
of charge and electric field intensity E has the dimensions of force. Let a 
charge of 1 coulomb be placed in an electric field whose intensity is 
1 volt/meter. 

,^ N . , , vv . volt ., joule . kilogram meter 

(10) 1 coulomb X 1 r = 1 + = 1 j 

x ' meter meter second 2 

The unit of force in the m.k.s. system is called the newton, and is equiva- 
lent to 1 joule/meter, or 10 5 dynes. 

The flux of the vector B shall be measured in webers, 

(11) 3> = f B n da webers, 

and the intensity of the field B, or flux density, may therefore be expressed 
in webers per square meter. According to (25), page 8, 

(12) f E .<to=-$ webei \ 3 . 

x ' Jc at second 

/*6 

The line integral I E ds is measured in volts and is usually called the 

electromotive force (abbreviated e.m.f.) between the points a and 6, 
although its value in a nonstationary field depends on the path of integra- 
tion. The induced e.m.f. around any closed contour C is, therefore, 
equal to the rate of decrease of flux threading that contour, so that 
between the units there exists the relation 

. ., ., weber 
1 volt = 



or 

,. i u i 3 u ^ e i kilogram meter 2 

(14) 1 weber = 1 - = 1 - ~ r - -r 
v ' ampere coulomb second 

It is important to note that the product of current and magnetic flux 
is an energy. Note also that the product of B and a velocity is measured 
in volts per meter, and is therefore a quantity of the same kind as E. 

/ K\ i ,- t T -i weber - kilogram 

(15) 1 unit of B = 1 75 = 1 - i T~ - -v 
v meter 2 coulomb second 

/i\ - ., - |T> , , , - weber vx - meter . volt - ., riT ,, 

(16) lumtof |B| |v| - I, X 1 ^^^^ = 1 jj^ = 1 unit of |E|. 



The units which have been deduced thus far constitute an absolute 
system in the sense that each has been expressed in term? of the four 



SBC. 1 8] THE M.K.S. OR GIORGI SYSTEM 21 

basic quantities, mass, length, time, and charge. That this system is 
identical with the practical series may be verified by the substitutions 

(17) 1 kilogram = 10* grams, 1 meter = 10 2 centimeters, 

1 coulomb = -nr abcoulomb. 

The numerical factors which now appear in each relation are observed 
to be those that relate the practical units to the absolute electromagnetic 
units. For example, from (6), 

1 , _- kilogram meter 2 __ 10 3 grams 10 4 centimeters 2 
coulomb 2 second 10~ 2 abcoulomb 2 seconds 

= 10 9 abohms; 
and again from (8), 

t u 1 kilogram -meter 2 ___ 10 3 grams 10 4 centimeters 2 
coulomb second 2 10" 1 abcoulomb second 2 

= 10 8 abvolts. 

The series must be completed by a determination of the units and 
dimensions of the vectors D and H. Since D = eE, H = - B, it is 

necessary and sufficient that o and /*o be determined such as to satisfy 
Eq. (2) and such that the proper ratio of practical to absolute units be 
maintained. We shall represent mass, length, time, and charge by the 
letters M, L, T 7 , and Q, respectively, and employ the customary symbol [A] 
as meaning "the dimensions of A." Then from Eq. (31), page 9, 

(20) f D n da = q coulombs 
and, hence, 

/ 91 \ mi _ coulombs Q 

(21) [D] - meter 2 = I? 

(22) [ co ] = r^-1 = C0ul mbs 

1 j L J 



- . 

# volt -meter ML 3 

The farad, a derived unit of capacity, is defined as the capacity of a 
conducting body whose potential will be raised 1 volt by a charge of 
1 coulomb. It is equal, in other words, to 1 coulomb/volt. The 
parameter CD in the m.k.s. system has dimensions, and may be measured 
in farads per meter. 

By analogy with the electrical case, the line integral f H ds taken 

Ja 

along a specified path is commonly called the magnetomotive force 



22 THE FIELD EQUATIONS [CHAP. I 

(abbreviated m.m.f.). In a stationary magnetic field 

(23) I H ds = J amperes, 

where / is the current determined by the flow of charge through any 
surface spanning the closed contour C. If tk** field is variable, I must 
include the displacement current as in (28), page 9. According to (23) 
a magnetomotive force has the dimensions of current. In practice, 
however, the current is frequently carried by the turns of a coil or winding 
which is linked by the contour C. If there are n such turns carrying a 
current I, the total current threading C is nl ampere-turns and it is 
customary to express magnetomotive force in these terms, although 
dimensionally n is a numeric. 

(24) [m.m.f.] = ampere-turns, 
whence 

/oc x rTT1 ampere-turns Q 

(25) IH] = meter = If' 

It will be observed that the dimensions of D and those of H divided by a 
velocity are identical. For the parameter MO we find 

/OA\ r i _ & volt * second ML 

(26) lMoJ "" 



ampere - meter ~ & 

As in the case of o it is convenient to express MO in terms of a derived 
unit, in this case the henry, defined as 1 volt-second/amp. (The henry 
is that inductance in which an induced e.m.f . of 1 volt is generated when 
the inducing current is varying at the rate of 1 amp. /sec.) The parameter 
Mo may, therefore, be measured in henrys per meter. 
From (22) and (26) it follows now that 



(27) 

LMOCO 

and hence that our system is indeed dimensionally consistent with 
Eq. (2). Since it is known that in the rationalized, absolute c.g.s. 
electromagnetic system MO is equal in magnitude to 4?r, Eq. (26) fixes also 
its magnitude in the m.k.s. system. 

/rtox . gram centimeters A 10~ 3 kilogram 10~ 2 meter 

(28) Mo = 4rr ~ r ; r-s = w ryrf * r~ 5 ; 

abcoulombs 2 10 2 coulombs 2 

or 

(29) N - 4r X 10-' kil0 g^'K o e 3 terS = 1-257 X 10- 



. 
coulombs 2 meter 



Sac. 1.0] VECTOR AND SCALAR POTENTIALS 23 

The appropriate value of o is then determined from 

(2) c = -^L= = 2.998 X 10 8 meters 

\AoMo second 

to be 



(30) eo - 8.854 X W-^ = 8 .854 X 10- 

3 



. 
kilogram meter 3 meter 

It is frequently convenient to know the reciprocal values of these factors, 



(31) = 0.7958 X 10 , = 0.1129 X 10 -, 
Mo henry c farad 

and the quantities 

(32) A p = 376.6 ohms, . p = 2.655 X 10~ 3 mho, 

\ *o \ MO 

recur constantly throughout the investigation of wave propagation. 

In Appendix I there will be found a summary of the units and dimen- 
sions of electromagnetic quantities in terms of mass, length, time, and 
charge. 

THE ELECTROMAGNETIC POTENTIALS 

1.9. Vector and Scalar Potentials. The analysis of an electromagnetic 
field is often facilitated by the use of auxiliary functions known as poten- 
tials. At every ordinary point of space, the field vectors satisfy the 
system 

(I) V X E + ~ = 0, (III) V B = 0, 
(II) VXH-~~ = J, (IV) V-D = p. 

According to (III) the field of the vector B is always solenoidal. Conse- 
quently B can be represented as the curl of another vector AO. 

(1) B = V X Ao. 

However A is not uniquely defined by (1) ; for B is equal also to the curl 
of some vector A, 

(2) B V X A, 

where 

(3) A = Ao - V^, 

&nd ^ is any arbitrary scalar function of position. 



24 THE FIELD EQUATIONS [CHAP. 1 

If now B is replaced in (I) by either (1) or (2), we obtain, respectively, 



Thus the fields of the vectors E H ~ and E + -rr are irrotational and 

ot ot 

equal to the gradients of two scalar functions < and <t>. 

(5) E = -V* - ^?, 

(6) E = -v* - 
The functions <f> and $ are obviously related by 

(7) *-* + & 

The functions A are vector potentials of the field, and the arc scalar 
potentials. AO and <o designate one specific pair of potentials from which 
the field can be derived through (1) and (5). An infinite number of 
potentials leading to the same field can then be constructed from (3) 
and (7). 

Let us suppose that the medium is homogeneous and isotropic, and 
that and p are independent of field intensity. 

(8) D = cE, B = M H. 

In terms of the potentials 



(9) x -, 

which upon substitution into (II) and (IV) give 

(10) V X V X A + ju*V -TJ + M* ^73- = ML 

All particular solutions of (10) and (11) lead to the same electromagnetic 
field when subjected to identical boundary conditions. They differ 
among themselves by the arbitrary function $. Let us impose now upon 
A and <t> the supplementary condition 

(12) V A + M ^r = 0. 

To do this it is only necessary that \l/ shall satisfy 
(13) 



SBC. 1.9] VECTOR AND SCALAR POTENTIALS 25 

where 4>o and A are particular solutions of (10) and (11). The potentials 
^ and A are now uniquely defined and are solutions of the equations 



(14) V X V X A - VV A + M - = /J, 

(15) 



Equation (14) reduces to the same form as (15) when use is made of 
the vector identity 

(16) V X V X A = VV A - V VA. 

The last term of (16) can be interpreted as the Laplacian operating on 
the rectangular components of A. In this case 

(17) ^ V 2 A- 

The expansion of the operator V VA in curvilinear systems will be 
discussed in Sec. 1.16, page 50. 

The relations (2) and (6) for the vectors B and E are by no means 
general. To them may be added any particular solution of the homo- 
geneous equations 

^T> 

(la) V X E + 25 = 0, (Ilia) V B = 0, 

(Ha) V X H - ~ = 0, (IVa) V-D = 0. 
ot 

From the symmetry of this system it is at once evident that it can be 
satisfied identically by 



(18) D = -V X A*, H - -V<f>* - -^, 
from which we construct 

(19) E = -J V X A*, B = -M (v<*>* + 
The new potentials are subject only to the conditions 



(20) 



26 THE FIELD EQUATIONS [CHAP. I 

A general solution of the inhomogeneous system (I) to (IV) is, therefore, 

(21) B = V X A fj, ~~^ M^0*, 

(22) E = -V0 - ~ - ~ V X A*, 

provided M and e are constant. 

The functions <* and A* are potentials of a source distribution which 
is entirely external to the region considered. Usually <* and A* are put 
equal to zero and the potentials of all charges, both distant and local, 
are represented by < and A. 

At any point where the charge and current densities are zero a 
possible field is <o = 0, A = 0. The function ^ is now any solution 
of the homogeneous equation 

(23) VV - M ^TJ = 0. 

Since at the same point the scalar potential satisfies the same equation, 
\l/ may be chosen such that <t> vanishes. In this case the field can be 
expressed in terms of a vector potential alone. 

(24) B = V X A, E = ~^> 

(25) V 2 A - /* -^ = 0, V A = 0. 

Concerning the units and dimensions of these new quantities, we 
note first that E is measured in volts/meter and that the scalar potential 
< is therefore to be measured in volts. If q is a charge measured in 
coulombs, it follows that the product q<f> represents an energy expressed 
in joules. From the relation B = V X A it is clear that the vector 
potential A may be expressed in webers/meter, but equally well in either 
volt-seconds/meter or in joules/ampere. The product of current and 
vector potential is therefore an energy. The dimensions of A* are found 
to be coulombs/meter, while <* will be measured in ampere-turns. 

1.10. Homogeneous Conducting Media. In view of the extreme 
brevity of the relaxation time it may be assumed that the density of 
free charge is always zero in the interior of a conductor. The field 
equations for a homogeneous, isotropic medium then reduce to 

(16) V X E + 22 = o, (III6) V B = 0, 

(116) VXH-^-<rE = 0, (IV6) V D = 0. 
ot 



SHC. 1.10] HOMOGENEOUS CONDUCTING MEDIA 27 

We are now free to express either B or D in terms of a vector potential. 
In the first alternative we have 

(26) B = V X A, E = -V* - ^ 

If the vector and scalar potentials are subjected to the relation 

(27) *-A + /* 



a possible electromagnetic field may be constructed from any pair of 
solutions of the equations 

(28) 
(29) 

w \jv 

As in the preceding paragraph one will note that the field vectors are 
invariant to changes in the potentials satisfying the relations 

/e .^ , , , d\l/ . . _ . 

(30) <b = </>o + -^7> A = Ao \y, 

ut 

where 6 , A are the potentials of a possible field and \l/ is an arbitrary 
scalar function. In order that A and <t> satisfy (27) it is only necessary 
that \l/ be subjected to the additional condition 

d^\b d\ls . $00 

(31) VnP /x IJLO" :== V AO i M^ ~r p>(T<f>Q, 

To a particular solution of (31) one is free to add any solution of the 
homogeneous equation 

Frequently it is convenient to choose \l/ such that the scalar potential 
vanishes. The field within the conductor is then determined by a single 
vector A. 

(33) B = V X A, E = -^ 

(34) V 2 A - /* ^ - iur ^ = 0, V A = 0. 

The field may also be defined in terms of potentials <* and A* by 

(35) D = -V X A*, H = -V<f>* - ^ - - A*. 



28 THE FIELD EQUATIONS [CHAP. I 

If <f>* and A* are to satisfy (28) and (29), it is necessary that they be 
related by 

(36) V A* + M 6 ? = 0. 

The field defined by (35) is invariant to all transformations of the poten- 
tials of the type 

(37) ** = tf + 1 + f **> A * = A * - v +*> 

where as above <* and A* are the potentials of any possible electro- 
magnetic field. To ensure the relation (36) it is only necessary that \l/* 
be chosen such as to satisfy 

(38) 

Finally, by a proper choice of \f/* the scalar potential <* may be made 
to vanish. 



(39) D=-VXA*, H= 

;)2A* ^A* 

(40) V'A* - M ^ - M<r ^ = 0, V A* = 0. 

1.11. The Hertz Vectors, or Polarization Potentials. We have seen 
that the integration of Maxwell's equations may be reduced to the 
determination of a vector and a scalar potential, which in homogeneous 
media satisfy one and the same differential equation. It was shown by 
Hertz 1 that it is possible under ordinary conditions to define an electro- 
magnetic field in terms of a single vector function. 

Let us confine ourselves for the present to regions of an isotropic, 
homogeneous medium within which there are neither conduction currents 
nor free charges. The field equations then reduce to the homogeneous 
system (la)-(IVa). We assume, for reasons which will become apparent, 
that the vector potential A is proportional to the time derivative of a 
vector n. 

/^i\ A dn 

(41) k== ^~df' 

Consequently, 

(42) B = MVX^ E= -V*-,*5, 

1 HBRTZ, Ann. Physik, 38, 1, 1888. The general solution is due to Righi: Boloyna 
Mem., (5) 9, 1, 1901, and II Nuovo Cimento, (5) 2, 2, 1901. 



SBC. 1.11] THE HERTZ VECTORS, OR POLARIZATION POTENTIALS 29 

and, when in turn this expression for E is introduced into (Ha), it is 
found that 

(43) | 

We recall that at points where there is no charge, the scalar function < 
is wholly arbitrary so long as it satisfies an equation such as (23). In 
the present instance it will be chosen such that 

(44) 4 = -V-n. 

Then upon integrating (43) with respect to the time, we obtain 



(45) VXVxn-VV-II + jLie -^ = constant. 

The particular value of the constant does not affect the determination of 
the field and we are therefore free to place it equal to zero. Equation 
(IVa) is also satisfied, for the divergence of the curl of any vector vanishes 
identically. Then we may state that every solution of the vector equation 

(46) vxv 

determines an electromagnetic field through 

(47) B = M eV X -~, E = VV . H - 

The condition that </> shall satisfy (23) is fulfilled in virtue of (46). One 
may replace (46) by 

(48) 



provided V 2 is understood to operate on the rectangular components of n. 
Since the vector D as well as B is solenoidal in a charge-free region, 
an alternative solution can be constructed of the form 

(49) A* = M e~; 0*=-V.n*, 

(50) D - -/16V X ^ H = VV II* - M -i 

where n* is any solution of (46) or (48). 

From these results we conclude that the electromagnetic field within 
a region throughout which c and ju are constant, p and J equal to zero, 
may be resolved into two partial fields, the one derived from the vector n 



30 THE FIELD EQUATIONS [CHAP. I 

and the other from the vector n*. The origin of these fields lies exterior 
to the region. To determine the physical significance of the Hertz 
vectors it is now necessary to relate them to their sources; in other words, 
we must find the inhomogeneous equations from which (48) is derived. 

Let us express the vector D in terms of E and the electric polariza- 
tion P. According to (6), page 11, D = E + P. Then in place of 
(Ha) and (IVa), we must now write 



It may be verified without difficulty that these two equations, as well 
as (la) and (Ilia), are still identically satisfied by (47), provided only 
that be replaced by and that n be now any solution of 

(52) 

The source of the vector H and the electromagnetic field derived from it is a 
distribution of electric polarization P. In due course we shall interpret 
the vector P as the electric dipole moment per unit volume of the medium. 
Since H is associated with a distribution of electric dipoles, the partial 
field which it defines is sometimes said to be of electric type, and n itself 
may be called the electric polarization potential. 

In like manner it can be shown that the field associated with n* is 
set up by a distribution of magnetic polarization. According to (6), 
page 11, the vector B is related to H by B = Mo(H + M), which when 
introduced into (la) and (Ilia) gives 

(53) VXE + M of~= -"'If' V ' H== ~ V ' M - 

Then these equations, as well as (Ha) and (IVa), are satisfied identically 
by (50) if we replace there /z by MO and prescribe that n* shall be a 
solution of 



(54) V 2 n*- M o6-^ r = -M. 

We shall show later that the polarization M may be interpreted as the 
density of a distribution of magnetic moment. The partial field derived 
from n* may be imagined to have its origin in magnetic dipoles and is 
said to be a field of magnetic type. 

The electric polarization P may be induced in the dielectric by the 
field E, but it may also contain a part whose magnitude is controlled 
by wholly external factors. In the practical application of the theory 
one is interested usually only in this independent part PC, which will be 



SBC. 1.11] THE HERTZ VECTORS, OR POLARIZATION POTENTIALS 31 

shown to represent the electric moment of dipole oscillators activated by 
external power sources. The same is true for the magnetic polarization. 
To represent these conditions we shall write (6), page 11, in the modified 
form 

(55) D = eE + Po, H = ~B - Mo, 

in which PO and Mo are prescribed and independent of E and H, and 
where the induced polarizations of the medium have again been absorbed 
into the parameters e and p. Then the electromagnetic field due to these 
distributions of PO and MO is determined by 

(56) E = VV - M*~ - MV X ~, 

(57) H = eVx 



when n and IT* are solutions of 



(58) VII - M = - Po, VII* - M e ~ = ~M . 

In virtue of the second of Eqs. (58) and of the identity (16) we may also 
write (57) as 

(59) H = eV X ~ + V X V X n* - Mo. 

ot 

Since B = V X A, it is evident from this last relation that the vector 
potential A may be derived from the Hertzian vectors by putting 

(60) A = M e ~ + M V X H* - Vft 

ot 

where \[/ is an arbitrary scalar function. The associated potential < is 

(61) 4,= _ v .n + ^, 

with $ subject only to the condition that it satisfy 

(62) W - ^ = 0. 

The extension of these equations to a homogeneous conducting 
medium follows without difficulty. The reader will verify by direct 
substitution that the system (I6)-(IV6), in a medium which is free of 



32 THE FIELD EQUATIONS [Caw. I 

fixed polarization P and M , is satisfied by 



(63) E = vxvxn- M vx 

(64) H = V X (* + rfl) + V X V X H*, 



TX vxn- vv-n + pe + nff = o, 
ffi.fi\ dt dt 

( ' a 2 n* an* 

v x v x n* - vv n* + ^ ^- + *w ^- = o. 

at* ot 

1.12. Complex Field Vectors and Potentials. It has been shown by 
Silberstein, Bateman, and others that the equations satisfied by the 
fields and potentials may be reduced to a particularly compact form by 
the construction of a complex vector whose real and imaginary parts are 
formed from the vectors defining the magnetic and electric fields. 1 The 
procedure has no apparent physical significance but frequently facilitates 
analysis. 

Consider again a homogeneous, isotropic medium in which D = dS, 
B = juH. If now we define Q as a complex field vector by 



(66) Q = B + 

the Maxwell equations (I)-(IV) reduce to 



(67) V X Q + n/^ = jj, V - Q = ip. 

The vector operation V X Q may be eliminated from (67) by the 
simple expedient of taking the curl of both members. By the identity 
(16) we obtain 



(68) VV - Q - V 2 Q + *V^V X - M V X J, 

ot 

which, on replacing the curl and divergence of Q by their values from 
(67), reduces to 

(69 ) 



When this last equation is resolved into its real and imaginary com- 

1 SILBERSTEIN, Ann. phys., 22, 24, 1907. Also Phil. Mag. (6) 23, 790, 1912. 
BATEMAN, "Electrical and Optical Wave Motion," Chap. I, Cambridge University 
Press. 



SEC. 1.12] COMPLEX FIELD VECTORS AND POTENTIALS 33 

ponents, one obtains the equations satisfied individually by the vectors 
E and H. 



(70) V *H- / = -VXJ, 

(71) **-,* = Mf + iv P . 

Next, let us define Q in terms of complex vector and scalar potentials 
L and 3> by the equation 

\T 

(72) Q = V X L - 



subject to the condition 

(73) VL + cM^f = 0. 

ot 

It will be verified without difficulty that (72) is an integral of (67) pro- 
vided the complex potentials satisfy the equations 



(74) V*L- eM = - M J, 

(75) 



If the real and imaginary parts of these potentials are written in the 
form 

(76) L = A - 

and substituted into (72), one finds again after separation of reals and 
imaginaries the general expressions for the field vectors deduced in 
Eqs. (21) and (22). 

If the free currents and charges are everywhere zero in the region 
under consideration, Eq. (67) reduces to 

The electromagnetic field may now be expressed in terms of a single 
complex Hertzian vector r. 



(78) Q = M cV X + iV^V X V X T, 

ot 

where T is any solution of 

(79) VT-e M f = 0. 



34 THE FIELD EQUATIONS [CHAP. I 

If, finally, r is defined as 

(80) r = n- 

and substituted into (78), one finds again after separation into real and 
imaginary parts exactly the expressions (47) and (50) for the electric 
and magnetic field vectors. 

When the medium is conducting, the field equations are no longer 
symmetrical and the method fails. The difficulty may be overcome 
if the field varies harmonically. The time then enters explicitly as a 
factor such as e i<at . After differentiating with respect to time, the 
system (I6)-(IV6) may be made symmetrical by introducing a complex 

inductive capacity e' = e i 

CO 

BOUNDARY CONDITIONS 

1.13. Discontinuities in the Field Vectors. The validity of the field 
equations has been postulated only for ordinary points of space; that is 
to say, for points in whose neighborhood the physical properties of the 
medium vary continuously. However, across any surface which bounds 
one body or medium from another there occur sharp changes in the 
parameters , p,, and a. On a macroscopic scale these changes may 
usually be considered discontinuous and hence the field vectors themselves 
may be expected to exhibit corresponding discontinuities. 

Let us imagine at the start that the surface S which bounds medium 
(1) from medium (2) has been replaced by a very thin transition layer 
within which the parameters e, /x, a vary rapidly but continuously from 
their values near S in (1) to their values near S in (2). Within this 
layer, as within the media (1) and (2), the field vectors and their first 
derivatives are continuous, bounded functions of position and time. 
Through the layer we now draw a small right cylinder, as indicated in 
Fig. 2a. The elements of the cylinder are normal to S and its ends lie 
in the surfaces of the layer so that they are separated by just the layer 
thickness A. Fixing our attention first on the field of the vector B, we 
have 

(1) < B n da = 0, 

when integrated over the walls and ends of the cylinder. If the base, 
whose area is Aa, is made sufficiently small, it may be assumed that B 
has a constant value over each end. Neglecting differentials of higher 
order we may approximate (1) by 

(2) (B ni + B n 2 )Aa + contributions of the walls = 0. 



SEC. 1.13] DISCONTINUITIES IN THE FIELD VECTORS 



35 



The contribution of the walls to the surface integral is directly pro- 
portional to AZ. Now let the transition layer shrink into the surface S. 
In the limit, as AZ > 0, the ends of the cylinder lie just on either side 
of S and the contribution from the walls becomes vanishingly small. 
The value of B at a point on S in medium (1) will be denoted by BI, while 




" Boundary 
surfaceS 



FIG. 2a. For the normal boundary condition. 



the corresponding value of B just across the surface in (2) will be denoted 
by B 2 . We shall also indicate the positive normal to S by a unit vector 
n drawn from (1) into (2). According to this convention medium (1) 
lies on the negative side of /S, medium (2) on the positive side, and 
HI = n. Then as AZ > 0, Aa 0, 



(3) 



(B 2 - BO n = 0; 



the transition of the normal component of B across any surface of discon- 
tinuity in the medium is continuous. Equation (3) is a direct consequence 
of the condition V B = 0, and is sometimes called the surface divergence. 




(1) Cy/^.lJi 

FIG. 26. For the tangential boundary condition. 

The vector D may be treated in the same manner, but in this case 
the surface integral of the normal component over a closed surface is 
equal to the total charge contained within it. 



(4) 



a) D n da = q. 



The charge is distributed throughout the transition layer with a den- 
sity p. As the ends of the cylinder shrink together, the total charge q 
remains constant, for it cannot be destroyed, and 



(5) 



q = p AZ Aa. 
0, the volume density p becomes infinite. It is then 



In the limit as AZ 

convenient to replace the product p AZ by a surface density w, defined as 



36 THE FIELD EQUATIONS [CHAP. I 

the charge per unit area. The transition of the normal component of the 
vector D across any surface >S is now given by 

(6) (D 2 - DO n = co. 

The presence of a layer of charge on S results in an abrupt change in the 
normal component of D, the amount of the discontinuity being equal to the 
surface density measured in coulombs per square meter. 

Turning now to the behavior of the tangential components we replace 
the cylinder of Fig. 2a by a rectangular path drawn as in Fig. 26. The 
sides of the rectangle of length As lie in either face of the transition layer 
and the ends which penetrate the layer are equal in length to its thick- 
ness AZ. This rectangle constitutes a contour Co about which 



(7) 



f E ds + f ^ - n da = 0, 

JCo JSo Ot 



where /So is the area of the rectangle and n its positive normal. The 
direction of this positive normal is determined, as in Fig. 1, page 8, by 
the direction of circulation about Co. Let TI and ^2 be unit vectors in 
the direction of circulation along the lower and upper sides of the rec- 
tangle as shown. Neglecting differentials of higher order, one may 
approximate (7) by 

3T> 

(8) (E ti + E * 2 ) As + contributions from ends = n As AZ. 

ut 

As the layer contracts to the surface $, the contributions from the seg- 
ments at the ends, which are proportional to AZ, become vanishingly 
small. If n is again the positive normal to S drawn from (1) into (2), 
we may define the unit tangent vector T by 

(9) * = n X n. 
Since 

(10) n X n E = n n X E, 
we have in the limit as AZ > 0, As > 0, 



(11) n n X (E 2 - Ei) + lim A - 0. 

L AZ-*0 \0l /I 

The orientation of the rectangle and hence also of no is entirely 
arbitrary, from which it follows that the bracket in (11) must equal 
zero, or 

(12) n X (E 2 - EO = - lim ^ AZ. 



SBC. 1.13] DISCONTINUITIES IN THE FIELD VECTORS 37 

The field vectors and their derivatives have been assumed to be bounded; 
consequently the right-hand side of (12) vanishes with AJ. 

(13) n X (Ei - Ei) = 0. 

The transition of the tangential components of the vector E through a surface 
of discontinuity is continuous. 

The behavior of H at the boundary may be deduced immediately 
from (12) and the field equation 

(14) I H ds I n da = I J n da. 

JCt JSg ut JS 9 

We have 

(15) n X (H 2 - Hi) = lim ( ~ + J J AZ. 

The first term on the right of (15) vanishes as AZ > because D and its 
derivatives are bounded. If the current density J is finite, the second 
term vanishes as well. It may happen, however, that the current 
J == J . n As AZ through the rectangle is squeezed into an infinitesimal 
layer on the surface S as the sides are brought together. It is con- 
venient to represent this surface current by a surface density K defined 
as the limit of the product J A I as Ai! > and J > o . Then 

(16) n X (H a - HO = K. 

When the conductivities of the contiguous media are finite, there can be 
no surface current, for E is bounded and hence the product 0-E AZ van- 
ishes with Ai. In this case, which is the usual one, 

(17) n X (H 2 HO = 0, (finite conductivity). 

Not infrequently, however, it is necessary to assume the conductivity 
of a body to be infinite in order to simplify the analysis of its field. One 
must then apply (16) as a boundary condition rather than (17). 

Summarizing, we are now able to supplement the field equations by 
four relations which determine the transition of an electromagnetic 
field from one medium to another separated by a surface of discontinuity. 

n (B 2 - Bi) = 0, n X (H, - HO = K, 
1 ; n X (E 2 - Ei) - 0, n (D 2 - DO = . 

From them follow immediately the conditions for the transition of the 
normal components of E and H. 

(19) 



n . ( Hl - ^H^ - (K n. (E, - SB^ - - 

\ M2 / \ 2 / f 



38 THE FIELD EQUATIONS [CHAP. I 

Likewise the tangential components of D and B must satisfy 



(20) n X D Z - Dx = 0, n X B 2 - ^Bi = M2 K. 

COORDINATE SYSTEMS 

1.14. Unitary and Reciprocal Vectors. It is one of the principal 
advantages of vector calculus that the equations defining properties 
common to all electromagnetic fields may be formulated without reference 
to any particular system of coordinates. To determine the peculiarities 
that distinguish a given field from all other possible fields, it becomes 
necessary, unfortunately, to resolve each vector equation into an equiva- 
.xmt scalar system in appropriate coordinates. 

In a given region let 

(1) u l = /i(x, y, z), w 2 = / 2 (z, y, z), u* = / 8 (x, y, z), 

be three independent, continuous, single-valued functions of the rec- 
tangular coordinates x. y t z. These equations may be solved with respect 
to x, y, z, and give 

(2) x = piCu 1 , w 2 , u 3 ), y = <p 2 <y, u 2 , w 3 ), z = <f>*(u\ u 2 , w 3 ), 

three functions which are also independent and continuous, and which 
are single- valued within certain limits. In general the functions <pi as 
well as the functions /i are continuously differentiable, but at certain 
singular points this property may fail and due care must be exercised in 
the application of general formulas. 

With each point P(x, y, z} in the region there is associated by means of 
(1) a triplet of values u l , u 2 , u 3 ] inversely (within limits depending on the 
boundaries of the region) there corresponds to each triplet u 1 , u 2 , u 3 a 
definite point. The functions u 1 , u 2 , u 3 are called general or curvilinear 
coordinates. Through each point P there pass three surfaces 

(3) u 1 = constant, u 2 = constant, u 3 = constant, 

called the coordinate surfaces. On each coordinate surface one coordi- 
nate is constant and two are variable. A surface will be designated by 
the coordinate which is constant. Two surfaces intersect in a curve, 
called a coordinate curve, along which two coordinates are constant and 
one is variable, A coordinate curve will be designated by the variable 
coordinate. 

Let r denote the vector from an arbitrary origin to a variable point 
P(x, y y z). The point, and consequently also its position vector r, may 
be considered functions of the curvilinear coordinates u 1 , u 2 , u 3 . 

(4) r = r(uS u\ u 3 ). 



SBC, 1.14] 



UNITARY AND RECIPROCAL VECTORS 



39 



A differential change in r due to small displacements along the coordinate 
curves is expressed by 



(5) 



dr = 

du 1 



Now if one moves unit distance along the i^-curve, the change in r is 
directed tangentially to this curve and is equal to dr/du 1 . The vectors 



(6) 



a 3 = 



dr 
du 3 ' 



are known as the unitary vectors associated with the point P. 
constitute a base system of reference for 
all other vectors associated with that 
particular point. 

(7) dr = ai du 1 + a 2 du* + a 8 du\ 

It must be carefully noted that the 
unitary vectors are not necessarily of unit 
length, and their dimensions will depend 
on the nature of the general coordinates. 
The three base vectors ai, a 2> a 3 de- 
fine a parallelepiped whose volume is 

(8) F = a! (a 2 X a 3 ) = a 2 (a 3 X aO 

= a 3 (ai X a 2 ). 

The three vectors of a new triplet defined by 



They 




FIQ. 3.- 



Base vectors for a curvilinear 
coordinate system. 



(9) 



a 2 = (a, X EI), 



a 5 = y (EI X a a ), 



are respectively perpendicular to the planes determined by the pairs 
(a 2 , a 3 ), (a s , ai), (EI, a 2 ). Upon forming all possible scalar products of 
the form E* E/, it is easy to see that they satisfy the condition 

(10) E"E,= tq, 

where $</ is a commonly used symbol denoting unity when i = j, and 
zero when i 9* j. The unitary vectors can be expressed in terms of the 
system E 1 , E 2 , E 8 by relations identical in form. 



(11) 



1_ 
V 



a 2 = (a 8 x a'), 



a, (a 1 X a 2 ). 



Any two sets of noncoplanar vectors related by the Eqs. (8) to (11) are 
said to constitute reciprocal systems. The triplefe a 1 , a 2 , a 8 are called 
reciprocal unitary vectors and they may serve as a base system quite 
as well as the unitary vectors themselves. 



40 THE FIELD EQUATIONS [CHAP. I 

If the reciprocal unitary vectors are employed as a base system, the 
differential dr will be written 

(12) dr = a 1 d^ + a 2 dw 2 + a 3 du*. 

The differentials du\, dw 2 , dw 3 are evidently components of dr in the direc- 
tions defined by the new base vectors. The quantities Wi, w 2 , w 3 are 
functions of the coordinates w 1 , w 2 , u s , but the differentials dui, dw 2 , duz 
are not necessarily perfect. On the contrary they are related to the 
differentials of the coordinates by a set of linear equations which in 
general are nonintegrable. Thus equating (7) and (12), we have 

(13) dr= 5)a*du= Ja'dw,. 



Upon scalar multiplication of (13) by a* and by a, in turn, we find, thanks 
to (10): 



3 3 



** o^ 

(14) du; = V a, a< dw*, du = V a* a' duj. 

-i A 

It is customary to represent the scalar products of the unitary vectors 
and those of the reciprocal unitary vectors by the symbols 



9ij = a a,- = 
(16) & = a . a/ = 



The components of dr in the unitary and in the reciprocal base systems 
are then related by 



3 3 

i- , 



3 

(17) duj = V 

i - 1 

A fixed vector F at the point P may be resolved into components 
either with respect to the base system ai, a 2> a 3 , or with respect to the 
reciprocal system a 1 , a 2 , a 8 . 



3 3 

(18) F = 



The components of F in the unitary system are evidently related to those 
in its reciprocal system by 



and in virtue of the orthogonality of the base veetors a,- with respect to 



SBC. 1.14J UNITARY AND RECIPROCAL VECTORS 41 

the reciprocal set a* as expressed by (10), we may also write 

It follows from this that (18) is equivalent to 

3 3 

(21) F = 



The quantities /* are said to be the contravariant components of the 
vector F, while the components // are called covariant. A small letter 
has been used to designate these components to avoid confusion with the 
components FI, F 2 , F$ of F with respect to a base system coinciding with 
the a^ but of unit length. It has been noted previously that the length 
and dimensions of the unitary vectors depend on the nature of the 
curvilinear coordinates. An appropriate set of unit vectors which, like 
the unitary set a*, are tangent to the t^'-curves, is defined by 

1 



t ij j > jj 3 -==. 

V011 V 922 V 033 

and, hence, 

(23) F = Fiii + F 2 i 2 + FA, 
with 

(24) F i 



The Fi are of the same dimensions as the vector F itself. 

The vector dr represents an infinitesimal displacement from the point 
P(w 1 , u 2 , w 3 ) to a neighboring point whose coordinates are u l + du l , 
u 1 + du 2 . u z + du*. The magnitude of this displacement, which con- 
stitutes a line element, we shall denote by ds. Then 

o O O 9 

(25) ds 2 = dr-di = V V ^.^du^dvf = 5) ] a* a' dm dw,-; 

ri yri 1 /-I 

or, in the notation of (15) and (16), 

3 3 

(26) ds 2 = V gaditdtf = ] gO'duidu*. 

tfZi i~i 

The t , and ^* J ' appear here as coefficients of two differential quadratic 
forms expressing the length of a line element in the space of the general 



42 



THE FIELD EQUATIONS 



[CHAP. I 



coordinates u { or of its reciprocal set w<. They are commonly called the 
metrical coefficients. 

It is now a relatively simple matter to obtain expressions for elements 
of arc, surface, and volume in a system of curvilinear coordinates. Let 
dsi be an infinitesimal displacement at P(u l , u 2 , u 3 ) along the t^-curve. 

(27) dsi = ai du 1 , d$i = \d$i\ = -\/0u du l . 
Similarly, for elements of length along the u 2 - and w 3 -curves, we have 

(28) ds<t = \/022 du*, d$ 3 = V033 du 9 . 

Consider next an infinitesimal parallelogram in the u ^surface bounded 

by intersecting u 2 - and w 3 -curves as 
indicated in Fig. 4. The area of such 
an element is equal in magnitude to 

(29) dai = |ds 2 X ds 3 | 

= |a 2 X a 3 | du 2 du 3 
= V(a 2 X a 3 ) (a 2 X a 3 ) du 2 du*. 

By a well-known vector identity 

(30) (a X b) (c X d) 

= (a-c)(b.d)- (a-d)(b.c), 

where a, b, c, d are any four vectors, 

FIG. 4. Element of area in the t^-surface. and hence 




(31) (a 2 X as) (a 2 X a 3 ) = (a 2 a 2 )(a 3 * a 3 ) - (a 2 a 3 )(a 3 a 2 ) 

= ^2 

For the area of an element in the i^-surface we have, therefore, 



= 022033 023- 



(32) Cfal = ^022033 - 023 du? du 3 , 

and similarly for elements in the w 2 - and w 3 -surfaces, 

(33) da 2 



U - 01i 



du 1 , 



da* = V0n022 - 012 du 1 du 2 . 
Finally, a volume element bounded by coordinate surfaces is written as 

(34) dv = dsi ds z X ds 8 = en a 2 X a f du 1 du 2 dw 3 . 
If now in (21) we let F = a 2 X a 3 , we have 

(35) a* X a, = (a 1 a, X a*^ + (a 2 a 2 X a 8 )a 2 + (a 3 a 2 X 



SEC. 1.14] UNITARY AND RECIPROCAL VECTORS 43 

or, on replacing the a* by their values from (8) and (9), 

(36) ai a 2 X a 3 = - a * Q [(a 2 X a 3 a 2 X a 3 )ai + 

**i * a 2 p\ a 3 

(a 3 X ai a 2 X a 3 )a 2 + (ai X a 2 a 2 X a 3 )a 8 ]. 

The quantities within parentheses can be expanded by (30) and the terms 
arranged in the form 

(37) (ai a 2 X a 3 ) 2 = ai ai[(a 2 a 2 )(a? a 3 ) - (a 2 a 3 )(a 3 a 2 )] 

+ ai a 2 [(a 2 a 3 )(a s ai) (a 2 ai)(a 3 a 3 )] 

+ ai a 3 [(a 2 ai)(a 3 a 2 ) - (a 2 a 2 )(a 3 aOJ. 

If finally the scalar products in (37) are replaced by their ga, we obtain 
as an expression for a volume element 



(38) dv = Vg du l du* du 3 , 

in which 

011 012 013 



(39) g = 



021 022 023 
031 032 033 



A corresponding set of expressions for the elements of arc, area, and 
volume in the reciprocal base system may be obtained by replacing the 
ga by the <y , but they will not be needed in what follows. 

Clearly the coefficients gy are sufficient to characterize completely 
the geometrical properties of space with respect to any curvilinear system 
of coordinates; it is therefore essential that we know how these coefficients 
may be determined. To unify our notation we shall represent the 
rectangular coordinates x t y, z of a point P by the letters x l , x 2 , x* respec- 
tively. Then 

(40) ds* = ((fa 1 ) 2 + (dx*Y + (dx*)\ 

In this most elementary of all systems the metrical coefficients are 

(41) ga = By, (da = 1, fc/ = when i 7* f). 

From the orthogonality of the coordinate planes and the definition (9), 
it is evident that, the unitary and the reciprocal unitary vectors are 
identical, are of unit length, and are the base vectors customarily repre- 
sented by the letters i, j, k. 

Suppose now that the rectangular coordinates are related functionally 
to a set of general coordinates as in (2) by the equations 

(42) x l x l (u l , u 2 , u*}, x* = x 2 (ttS w 2 , w 8 )> z 3 = x*(u l , w 2 , w s ). 



44 THE FIELD EQUATIONS [CHAP. I 

The differentials of the rectangular coordinates are linear functions of the 
differentials of the general coordinates, as we see upon differentiating 
Eqs. (42). 



. 

du 1 du* 

r)r 2 

(43) 



According to (26) and (40) 
(44) ds s = V 5) 



whence on squaring the differentials in (43) and equating coefficients 
of like terms we obtain 



ii ~ du 1 du* du* du> du* du! 

1.15. The Differential Operators. The gradient of a scalar function 
^(w 1 , u i / -u 3 ) is a fixed vector defined in direction and magnitude as the 
maximum rate of change of < with respect to the cordinates. The 
variation in incurred during a displacement dr is, therefore, 



(46) d* = V* dt = du< - 

tl 

Now the dw* are the contravariant components of the displacement vector 
dr, and hence by (20) , 

(47) du { = a dr. 
This value for du { introduced into (46) leads to 



and, since the displacement dr is arbitrary, we find for the gradient of a 
scalar function in any system of curvilinear coordinates: 



() ^ 

In this expression the reciprocal unitary vectors constitute the base 



SBC. 1.15] 



THE DIFFERENTIAL OPERATORS 



45 



system, but these may be replaced by the unitary vectors through the 
transformation 

3 

(50) 

The divergence of a vector function F(u 1 , u 2 , u z ) at the point P may 
be deduced most easily from its definition in Eq. (9), page 4, as the 
limit of a surface integral of the normal component of F over a closed 
surface, per unit of enclosed volume. Consider those two ends of the 
volume element illustrated in Fig. 5 which 
lie in i6 2 -surfaces. The left end is located 
at u 2 , the right at u 2 + du 2 . The area 
of the face at u 2 is (ai X &z)du l du*, the 
order of the vectors being such that 
the normal is directed outward, i.e., to 
the left. The net contribution of these 
two ends to the outward flux is, therefore, 

(51) [F (as X ai) du 1 du*] u *+ du * 

+ [F (ai X a 3 ) du 1 du*] v *, 

the subscripts to the brackets indicating 
that the enclosed quantities are to be 
evaluated at u 2 + du 2 and u 2 respective- 
ly. For sufficiently small values of du 2 , 

(51) may be approximated by the linear term of a Taylor expansion, 

(52) ~ (F - a 3 X en du 1 du 2 du*), 

ai X a 3 having been replaced by a 3 X ai. Now by (21), (20), and 
(37) we have 

(53) F a 3 x at = F a 2 (a 2 a 3 X aO = f 2 Vg; 
hence the contribution of the two ends to the surface integral is 

a 




FIG. 5. Element of volume in 
curvilinear coordinate system. 



(54) 



du 2 



(f 2 Vg) du* du 2 



Analogous contributions result from the two remaining pairs of faces. 
These are to be measured per unit volume; hence we divide by dv = -\/g 
du 1 du 2 du 3 and pass to the limit du 1 * 0, du 2 > 0, du 3 0, ensuring 
thereby the vanishing of all but the linear terms in the Taylor expansion. 
The divergence of a vector F referred to a system of curvilinear coordi- 
nates is, therefore, 



(55) 



V F = 



t = l 



46 



THE FIELD EQUATIONS 



[CHAP. I 



The curl of the vector F is found in the same manner by calculating 
the line integral of F around an infinitesimal closed path. According 
to Eq. (21), page 7, the component of the curl in a direction defined 
by a unit normal n is 



(56) 



(V X F) n 



= lim = I 

c-o & Jc 



F-ds. 



Let us take the line integral of F about the contour of a rectangular 
element of area located in the ^-surface, as indicated in Fig. 6. The 
sides of the rectangle are a 2 du 2 and a 3 du 3 . The direction of circulation 

is such that the positive normal is in the 
sense of the positive i^-curve. The con- 
tribution from the sides parallel to w 3 -curves 
is 




(F a 3 



* (F a 3 



from the bottom and top parallel to 
curves, we obtain 



- (F a 2 



(F a 2 



FIG. 6. Calculation of the curl 
in curvilinear coordinates. 



Approximating these differences by the 
linear terms of a Taylor expansion, we 
obtain for the line integral 



- 5? CP-.O] 



du 2 du 3 . 



This quantity must now be divided by the area of the rectangle, or 
\/(a2 X a 3 ) (a 2 X a 3 ) du 2 du 3 . As for the unit normal n, we note that 
the reciprocal vector a 1 , not the unitary vector ai, is always normal to the 
^-surface. Its magnitude must be unity; hence 



(58) 



n = 



These values introduced into (56) now lead to 



(59) (V X F) 



1 



V (a 2 X as) (a 2 X 



By (9) and f37) 

(60) a 2 X a, = [a t (a 2 X 



a 1 ; 



SEC. 1.16] ORTHOGONAL SYSTEMS 47 

hence (59) reduces to 

(61) (vx^.a 



The two remaining components of V X F are obtained from (61) by 
permutation of indices. Then by (21) 



(62) 



VXF=Y(VXF- a*)*. 

i^l 



Remembering that F a* is the covariant component /, we have for the 
curl of a vector with respect to a set of general coordinates 



+ 



(-!&)*] 



Finally, we consider the operation V 2 <, by which we must understand 
V V<t>. We need only let F = V<f> in (55). The contravariant com- 
ponents of the gradient are 

(64) p = F a* = 

j-r 
Then 

(65) V V<t> = 



1.16. Orthogonal Systems. Thus far no restriction has been imposed 
on the base vectors other than that they shall be noncoplanar. Now it 
happens that in almost all cases only the orthogonal systems can be 
usefully applied, and these allow a considerable simplification of the 
formulas derived above. Oblique systems might well be of the greatest 
practical importance; but they lead, unfortunately, to partial differential 
equations which cannot be mastered by present-day analysis. 

The unitary vectors ai, a 2 , a 3 of an orthogonal system are by definition 
mutually perpendicular, whence it follows that a* is parallel to a and 
is its reciprocal in magnitude. 

(66) a --i-a = la. 

a a 0u 

Furthermore 

(67) ai a 2 = a 2 a s = a 8 ai 0; 



48 THE FIELD EQUATIONS [CHAP. I 

hence ga = 0, when i ^ j. It is customary in this orthogonal case to 
introduce the abbreviations 

(68) hi = 



The hi may be calculated from the formula 



at* 

(70) 



J , 



although their value is usually obvious from the geometry of the system. 
The elementary cell bounded by coordinate surfaces is now a rectangular 
box whose edges are 



(71) dsi = hi du l , ds% = h% dw 2 , ds* = h$ du 3 , 
and whose volume is 

(72) dv = hihji* du l du* du*. 

All off-diagonal terms of the determinant for g vanish and hence 

(73) Vg = hihji*. 

The distinction between the contravariant and covariant components 
of a vector with respect to a unitary or reciprocal unitary base system is 
essential to an understanding of the invariant properties of the differential 
operators and of scalar and vector products. However, in a fixed refer- 
ence system this distinction may usually be ignored. It is then con- 
venient to express the vector F in terms of its components, or projections, 
Fi 9 ^2, ^3 on an orthogonal base system of unit vectors ii, i 2) is. By (22) 
and (66) 

(74) a< = W, a = ~i<. 

Al 

In terms of the components JP the contravariant and covariant com- 
ponents are 

(75) /' = r F '' /< = ***. 

'* 

Also 

(76) F = Fiii + F 2 i 2 + F 8 i 8> 

(77) ij i k = 5, fc . 

The gradient, divergence, curl, and Laplacian in an orthogonal system 
of curvilinear coordinates can now be written down directly from the 
results of the previous section. 



SEC. 1.16] ORTHOGONAL SYSTEMS 

From (49) we have for the gradient 

(78) V<t> = 



49 



j-i 
According to (55) the divergence of a vector F is 

(79) v ' F = d^[^ (wo + ^ 

For the curl of F we have by (63) 
V X F = T-i 



It may be remarked that (80) is the expansion of the determinant 



(81) 



V X F = 



d d d 

du l du 2 du* 
Fi hzFz 



Finally, the Laplacian of an invariant scalar <j> is 



By an invariant scalar is meant a quantity such as temperature or 
energy which is invariant to a rotation of the coordinate system. The 
components, or measure numbers, JF\ of a vector F are scalars, but they 
transform with a transformation of the base vectors. Now in the 
analysis of the field we encounter frequently the operation 



(83) 



V X V X F = VV F - V VF. 



No meaning has been attributed as yet to V VF. In a rectangular, 
Cartesian system of coordinates x l , x 2 , # 3 , it is clear that this operation is 
equivalent to 



(84, 




i.e., the Laplacian acting on the rectangular components of F, In genet- 



50 THE FIELD EQUATIONS [CHAP. I 

alized coordinates V X V X F is represented by the determinant 

ii 

/i 2 /i3 

A 
(85) V X V X F = 



a 

du* 

A.L 



IT 1 - 

/Il/l2 

d 

du z 



du l 



The vector V VF may now be obtained by subtraction of (85) from the 
expansion of W F, and the result differs from that which follows 
a direct application of the Laplacian operator to the curvilinear com- 
ponents of F. 

1.17. The Field Equations in General Orthogonal Coordinates. In 
any orthogonal system of curvilinear coordinates characterized by the 
coefficients /ti, 7i 2 , /is, the Maxwell equations can be resolved into a set of 
eight partial differential equations relating the scalar components of the 
field vectors. 



1 



(D 



^ (h*E z ) - - 2 (hjlj 



du* 

a 



a#i 
a* 



= 0. 



= o. 



~ 3 = 0. 



CrjL/l * 

~ ~aT = - 71 - 



et " z - 
~ "aT = /8 ' 



(A^BO + (WA) + (MA) = 0. 



_3_ 

dM 1 

a 



(IV) 



+ 






+ ^, 



Sac, 1.18] PROPERTIES OF SOME ELEMENTARY SYSTEMS 



51 



It is not feasible to solve this system simultaneously in such a manner 
as to separate the components of the field vectors and to obtain equations 
satisfied by each individually. In any given problem one must make the 
most of whatever advantages and peculiarities the various coordinate 
systems have to offer. 

1.18. Properties of Some Elementary Systems, An orthogonal 
coordinate system has been shown to be completely characterized by the 
three metrical coefficients, hi, hi, A 3 . These parameters will now be 
determined for certain elementary systems and in a few cases the differ- 
ential operators set down for convenient 
reference. 

1. Cylindrical Coordinates. Let P f be 
the projection of a point P(x, y, z) on the 
z-plane and r, be the polar coordin- 
ates of P' in this plane (Fig. 7). The 
variables 



(86) u l 



u 3 z, 



are called circular cylindrical coordin- 
ates. They are related to the rectan- 
gular coordinates by the equations 




Fl ' 



(87) 



x = r cos 0, 



z = z. 



y = r sin 0, 

The coordinate surfaces are coaxial cylinders of circular cross section 
intersected orthogonally by the planes 6 = constant and z = constant. 
The infinitesimal line element is 

(88) ds 2 = dr 2 + r 2 dP + dz\ 
whence it is apparent that the metrical coefficients are 

(89) hi = 1, h, = r, A 8 = 1. 
If ^ is any scalar and F a vector function we find: 



VTJ x v 
jp = - | 



<9o) 



rdr dr 



r 2 60* 



52 THE FIELD EQUATIONS [CHAP. I 

2. Spherical Coordinates. The variables 

(91) u l = r, w 2 = 6, u* *= <f>, 
related to the rectangular coordinates by the transformation 

(92) x = r sin cos <, y = r sin sin <, 2 = r cos 0, 

are called the spherical coordinates of the point P. The coordinate 
surfaces, r = constant, are concentric spheres intersected by meridian 
planes, <t> = constant, and a family of cones, 6 = constant. The unit 

vectors ii, i 2 , is are drawn in the direc- 
tion of increasing r, 0, and <t> such as to 
constitute a right-hand base system, as 
indicated in Fig. 8. The line element 
is 

(93) ds* = dr 2 + r 2 dO* 

+ r 2 sin 2 d<t>*, 

whence for the metrical coefficients we 
obtain 




FIQ. 8. Spherical coordinates. 



(94) hi = 1, A 2 
These values lead to 



= r, 



As = r sin 6. 



(95) 




r 2 sin 2 d<t> 2 



3. Elliptic Coordinates. Let two fixed points Pi and P z be located at 
z = c and x = c on the x-axis and let r\ and rj be the distances of a 
variable point P in the z-plane from Pi and P 2 . Then the variables 



(96) t 

defined by equations 
(97) 



* = z. 



2c 



SEC. 1.18] PROPERTIES OF SOME ELEMENTARY SYSTEMS 53 

are called elliptic coordinates. From these relations it is evident that 

(98) | 1, -1 SS if ^ 1. 

The coordinate surface, = constant, is a cylinder of elliptic cross 
section, whose foci are PI and P* The semimajor and semiminor axes 
of an ellipse are given by 

(99) a = c, 
and the eccentricity is 
(100) 



_ c _ 1 
e " a " I " 



The surfaces, 77 = constant, represent a family of confocal hyperbolic 




V-TT 

FIQ. 9. Coordinates of the elliptic cylinder. Ambiguity of sign is avoided by placing 

cosh u, q cos v. 

cylinders of two sheets as illustrated in Fig. 9. The equations of these 
two confocal systems are 

r 

> 



T* r/ 2 r* 

(ioi) fi + ^n = ", f-, 



i -u 2 



from which we deduce the transformation 
(102) 



x = 



= c 



The variable ij corresponds to the cosine of an angle measured from the 
s-axis and the unit vectors ii, i a of a right-hand base system are therefore 



THE FIELD EQUATIONS 



[CHAP. I 



drawn as indicated in Fig. 9, with is normal to the page and directed 
from the reader. 

The metrical coefficients are calculated from (102) and (70), giving 



(103) Ai = 

4. Parabolic Coordinates. If r, 6 are polar coordinates of a variable 
point in the z-plane, one may define two mutually orthogonal families 
of parabolas by the equations 



(104) 



^ / ^ A 

\/2r sin - r, = \/2r cos - 



The surfaces, = constant and n constant, are intersecting parabolic 



$=5 



>7=5 




45 

FIQ. 10. Parabolic coordinates. 

cylinders whose elements are parallel to the z-axis as shown in Fig. 10. 
The parameters 

(105) u l = , w 2 = 17, w 8 = -z 

are called parabolic coordinates. Upon replacing r and in (104) by 
rectangular coordinates we find 



SEC. 1.18] PROPERTIES OF SOME ELEMENTARY SYSTEMS 



55 



whence for the transformation from rectangular to parabolic coordinates 
we have 



(107) 



x = 



y = 



z = z. 



The unit vectors ii and i 2 are directed as shown in Fig. 10, with I 3 normal 
to the page and away from the reader. The calculation of the metrical 
coefficients from (107) and (70) leads to 



(108) 



7*3= 1. 



5. Bipolar Coordinates. Let PI and P 2 be two fixed points in any 
z-plane with the coordinates (a, 0), ( a, 0) respectively. If $ is a 
parameter, the equation 

(109) (x - a coth ) 2 + y 2 = a 2 csch 2 , 

describes two families of circles whose centers lie on the x-axis. These 
two families are symmetrical with respect to the T/-axis as shown in 




FIQ. 11. Bipolar coordinates. 



Fig. 11. The point P t at (a, 0) corresponds to = +00, whereas its 
image P 2 at (a, 0) is approached when = oo . The locus of (109), 
when f = 0, coincides with the 7/-axis. The orthogonal set is likewise 
a family of circles whose centers all lie on the #-axis and all of which pass 
through the fixed points Pi and P 2 . They are defined by the equation, 

(110) x 2 + (y - a cot 17) 2 = a 2 esc 2 1?, 



56 THE FIELD EQUATIONS [CHAP. I 

wherein the parameter 17 is confined to the range ^ 77 g 2r. In order 
that the coordinates of a point P in a given quadrant shall be single- 
valued, each circle of this family is separated into two segments by the 
points PI and P*. A value less than TT is assigned to the arc above the 
x-axis, while the lower arc is denoted by a value of 77 equal to TT plus 
the value of 17 assigned to the upper segment of the same circle. 
The variables 

(111) u l = , u 2 = 77, n 3 = z, 

are called bipolar coordinates. From (109) and (110) the transformation 
to rectangular coordinates is found to be 

/nox a sinh a sin 77 

(112) x = r-r 9 y = T-T > z = z. 

cosh cos TJ cosh cos 17 

The unit vectors ii and i 2 are in the direction of increasing and 17 as 
indicated in Fig. 11, while i 3 is directed away from the reader along the 
z-axis. The calculation of the metrical coefficients yields 

(113) hi = h 2 = r--^ , A, = 1. 

v ' cosh cos 77 

6. Spheroidal Coordinates. The coordinates of the elliptic cylinder 
were generated by translating a system of confocal ellipses along the 
z-axis. The spheroidal coordinates are obtained by rotation of the 
ellipses about an axis of symmetry. Two cases are to be distinguished, 
according to whether the rotation takes place about the major or about 
the minor axis. In Fig. 9 the major axes are oriented along the x-axis of 
a rectangular system. If the figure is rotated about this axis, a set of 
confocal prolate spheroids is generated whose orthogonal surfaces are 
hyperboloids of two sheets. If <t> measures the angle of rotation from 
the 2/-axis in the x-plane and r the perpendicular distance of a point 
from the x-axis, so that 

(114) y = r cos 0, z = r sin 0, 
then the variables 

(115) u^ = fc u* = 17, u* = *, 

defined by (97) and (114) are called prolate spheroidal coordinates. In 
place of (101) we have for the equations of the two confocal systems 



Sue. 1.18] PROPERTIES OF SOME ELEMENTARY SYSTEMS 57 

from which we deduce 
(117) x = eft, y = 



- 1)(1 - rrOsin 

(118) ^ 1, -1 g r? g 1, ^ g 27T. 

A calculation of the metrical coefficients gives 

(119) A 1 = 




When the ellipses of Fig. 9 are rotated about the y-axis, the spheroids 
are oblate and the focal points PI, P 2 describe a circle in the plane y = 0. 
Let r, 0, y, be cylindrical coordinates about the 2/-axis, 

(120) z = r cos <, x = r sin <. 

If by PI and P 2 we now understand the points where the focal ring of 
radius c intercepts the plane 4> = constant, the variables and t\ are 
still defined by (97) ; but for the equations of the coordinate surfaces we 
have 



from which we deduce the transformation from oblate spheroidal 
coordinates 

(122) u 1 = f, u 2 = T?, ti 3 = 0, 

to rectangular coordinates 



(123) x = c^ sin 0, t/ = c V(f 2 - 1)(1 - r; 2 ), z = cf iy cos 0. 

The surfaces, = constant, are oblate spheroids, whereas the orthogonal 
family, rj = constant, are hyperboloids of one sheet. The metrical 
coefficients are 




(124) *i = <ufe r' ^ = c 



The practical utility of spheroidal coordinates may be surmised from 
the fact that as the eccentricity approaches unity the prolate spheroids 
become rod-shaped, whereas the oblate spheroids degenerate into flat, 
elliptic disks. In the limit, as the focal distance 2c and the eccentricity 
approach zero, the spheroidal coordinates go over into spherical coordin- 
ates, with > r, rj > cos 0. 

7. Pardboloidal Coordinates. Another set of rotational coordinates 
may be obtained by rotating the parabolas of Fig. 10 about their axis 



58 THE FIELD EQUATIONS [CHAP. I 

of symmetry. The variables 

(125) u l = $, u 2 = T], w 8 = 4>, 
defined by 

(126) x = (ft cos <#>, 2/ = (to sin *> 2 = *( 2 ~ ^ 2 ) 

are called paraboloidal coordinates. The surfaces, = constant, 
rj = constant, are paraboloids of revolution about an axis of symmetry 
which in this case has been taken coincident with the z-axis. The plane, 
y = 0, is cut by these surfaces along the curves 



- *), ** = V (^ + *)> 



(127) * = 

which are evidently parabolas whose foci are located at the origin and 
whose parameters are 2 and 7j 2 . The metrical coefficients are 

(128) hi = A 2 - VlFTT 2 , >*3 = {17. 
8. Ellipsoidal Coordinates. The equation 

(129) JT. + S + ?- 1 ' ( >6>C) ' 
is that of an ellipsoid whose semiprincipal axes are of length a, 6, c. Then 



(130) + - + -1, (-e. >,>->>, 



o^Tf ^6 2 + f ' c 2 + f -' v 

are the equations respectively of an ellipsoid, a hyperboloid of one sheet, 
and a hyperboloid of two sheets, all confocal with the ellipsoid (129). 
Through each point of space there will pass just one surface of each kind, 
and to each point there will correspond a unique set of values for , ?/, f . 
The variables 
(131) u* = *, u 2 = n, u* = f, 

are called ellipsoidal coordinates. The surface, % = constant, is a 
hyperboloid of one sheet and t\ = constant, a hyperboloid of two sheets. 
The transformation to rectangular coordinates is obtained by solving 



SEC. 1.19] ORTHOGONAL TRANSFORMATIONS 

(130) simultaneously for x, y, z. This gives 



59 



(132) 



x = 
</= 
*= 



- 2 



a 2 ) 



(c 2 - 6 2 )(a 2 - 6 2 ) 
"(a 2 - c 2 )(6 2 ^ =: 7 2 y" 



The mutual orthogonality of the three families of surfaces may be verified 
by calculating the coefficients ga from (132) by means of (45). They are 
zero when i ^ j; for the diagonal terms we find 



(133) 



^3 ^ 



( - u)( - r) 



(r - )(r - 



It is convenient to introduce the abbreviation 

(134) R 9 * V(s + <* 2 )( s + W)(s + i 

For the Laplacian of a scalar \l/ we then have 

4 



( = 



(135) 



o 



THE FIELD TENSORS 

1.19. Orthogonal Transformations and Their Invariants. In the 

theory of relativity one undertakes the formulation of the laws of physics, 
and in particular the equations of the electromagnetic field, such that 
they are invariant to transformations of the system of reference. 
Although in the present volume we shall have no occasion to examine the 
foundations of the relativity theory, it will nevertheless prove occasion- 
ally advantageous to employ the symmetrical, four-dimensional notation 
introduced by Minkowski and Sommerfeld and to deduce the Lorentz 
transformation with respect to which the field equations are invariant. 
To discover quantities which are invariant to a transformation from one 
system of general curvilinear coordinates to another, it is essential that 



60 THE FIELD EQUATIONS [CHAP. I 

one distinguish between the covariant and contravariant components 
of vectors and between unitary and reciprocal unitary base systems. 
For our present purposes it will be sufficient, however, to confine the 
discussion to systems of rectangular, Cartesian coordinates in which, as 
we have seen, covariant and contravariant components are identical. 1 

Let ii, i 2 , ia be three orthogonal, unit base vectors defining a rectangu- 
lar coordinate system X whose origin is located at the fixed point 0, and 
let r be the position vector of any point P with respect to 0. 



(1) r = xiii + a? 8 i 2 + 
and since 

(2) iy U /*, 
the coordinates of P in the system X are 

(3) Xk = r i*. 

Suppose now that ij, ij, i are the base vectors of a second rectangular 
system X' whose origin coincides with and which, therefore, differs 
from X only by a rotation of the coordinate axes. Since 

(4) r = x(i{ + x'& + xjij, 
the coordinates of P with respect to X' are 

(5) zj = r ij = xiii ij + z 2 i 2 % + z 3 i 3 i}; 



each coordinate of P in X' is a linear function of its coordinates in X, 
whereby the coefficients 

(6) Oik = ij U 

of the linear form are clearly the direction cosines of the coordinate axes 
of X' with respect to the axes of X. A rotation of a rectangular coordi- 
nate system effects a change in the coordinates of a point which may be 
represented by the linear transformation 

(7) *} - ]g <^*> (3 = *> 2 > 3 )- 

Jk-l 

The coefficients a# are subject to certain conditions which are a 
consequence of the fact that the distance from to P, that is to say, the 

l This section is based essentially on the following papers: MINKOWSKI, Ann. 
Phyaik, 47, 927, 1915; SOMMERFELD, Ann. Physik, 32, 749, 1910 and 33, 649, 1910; 
MIE, Ann. Physik, 37, 511, 1912; PAULI, Relativitatstheorie, in the Encyklopadie der 
mathematischen Wissenschaften, Vol. V, part 2, p. 539, 1920. 



SEC. 1.19] ORTHOGONAL TRANSFORMATIONS 61 

magnitude of r, is independent of the orientation of the coordinate system. 



3 3 / 3 \/ 3 - 33 

(Q\ V (~'\2 =- V / V fl T I I V 

W ( 2j wJ - .2^ \2^ a x *) \Zf^ 

whence it follows that 

(10) V...., - A. - *> wheni = fc > 



Equation (10) expresses in fact the relations which must prevail among 
the cosines of the angles between coordinate axes in order that they be 
rectangular and which are, therefore, known as conditions of orthog- 
onality. The system (7), when subject to (10), is likewise called an 
orthogonal transformation. As a direct consequence of (10), it may be 
shown that the square of the determinant |a#| is equal to unity and hence 
\a>]k\ = 1. Any set of coefficients a,- fc which satisfy (10) define an 
orthogonal transformation in the sense that the relation (8) is preserved. 
Geometrically the transformation (7) represents a rotation only when 
the determinant |a 3 jfc| = +1. The orthogonal transformation whose 
determinant is equal to 1 corresponds to an inversion followed by a 
rotation. 

Since the determinant of an orthogonal transformation does not 
vanish, the Xk may be expressed as linear functions of the Xy. These 
relations are obtained most simply by writing as in (5) : 

(11) Xk = r U = x(i( I* + z& ifc + ^ij - i*, 
or 

3 

(12) x k - ]g a,-**}, (* = 1, 2, 3), 

j'-i 

whence it follows from (8) that 

3 

(13) 



Let A be any fixed vector in space, so that 

3 3 

(14) A = J A k i k = V A&. 

* - 1 * - 1 

The component Ay of this vector with respect to the system X' is given by 

(15) A;. = A i} = V ^* ' *J = 2 



62 THE FIELD EQUATIONS [CHAP. I 

thus the rectangular components of a fixed vector upon rotation of the 
coordinate system transform like the coordinates of a point. Now 
while every vector has in general three scalar components, it does not 
follow that any three scalar quantities constitute the components of a 
vector. In order that three scalar s AI, A 2, A 3 may be interpreted as the 
components of a vector, it is necessary that they transform like the coordinates 
of a point. 

Among scalar quantities one must distinguish the variant from the 
invariant Quantities such as temperature, pressure, work, and the liko 
are independent of the orientation of the coordinate system and are, 
therefore, called invariant scalars. On the other hand the coordinates 
of a point, and the measure numbers, or components, of a vector have 
only magnitude, but they transform with the coordinate system itself. 
We know that the product A B of two vectors A and B is a scalar, but a 
scalar of what kind? In virtue of (12) and (13) we have 

3 3 / 3 \/ 3 \ 3 

(16) A-B = 2 AkBk = 2(2 a '* A 0(S a* B '<) = 2 A 'i B 'i> 

the scalar product of two vectors is invariant to an orthogonal transformation 
of the coordinate system. 

Let < be an invariant scalar and consider the triplet of quantities 

(17) B t = J, ( = 1, 2, 3). 

Now by (12), 

(18) ~, = a^, 
and hence 

(19) * = - 



1 

the Bi transform like the components of a vector and therefore the gradient of 4 , 



calculated at a point P, is a fixed vector associated with that point. 
Let Ai be a rectangular component of a vector A, and 



(21) B, = 
Then by (18) and (15), 

(22) l* 



SBC. 1.19] ORTHOGONAL TRANSFORMATIONS 63 

whence, from (10), it follows that 



the divergence of a vector is invariant to an orthogonal transformation of the 
coordinate system. 

Lastly, since the gradient of an invariant scalar is a vector and since 
the divergence of a vector is invariant, it follows that the Laplacian 

(24) V V = V . V 4, 






is invariant to an orthogonal transformation. 

The transformation properties of vectors may be extended to mani- 
folds of more than three dimensions. Let &i, & 2 , 3, x * be the rectangular 
coordinates of a point P with respect to a reference system X in a four- 
dimensional continuum. The location of P with respect to a fixed origin 
is determined by the vector 



(25) r = 

/=- 



The linear transformation 

(26) *J = , a,***, (j = 1, 2, 3, 4), 



= 1 
will be called orthogonal if the coefficients satisfy the conditions 

4 
f f >'7\ X^ < 

\*<) 2if a )> a i k = *** 

J-l 

The characteristic property of an orthogonal transformation is that it 
leaves the sum of the squares of the coordinates invariant: 

(28) 

y^i 

The square of the determinant formed from the a,-* is readily shown to be 
positive and equal to unity, and hence the determinant itself may equal 
1. However, if (26) is to include the identical transformation 

(j = 1, 2, 3, 4), 

it is obvious that the determinant must be positive. Henceforth we 
shall confine ourselves to the subgroup of orthogonal transformation? 



64 THE FIELD EQUATIONS [CHAP. I 

characterized by (27) and the condition 
(30) M = +1. 

The transformation then corresponds geometrically to a rotation of the 
coordinate axes. 

A four-vector is now defined as any set of four variant scalars 
Ai (i = 1, 2, 3, 4) which transform with a rotation of the coordinate 
system like the coordinates of a point. 



(31) 



It is then easy to show, as above, that the scalar product of two four- 
vectors and the four-dimensional divergence of a four-vector are invariant 
to a rotation of the coordinate system. 

4 4 

(32) A B = "SJ A k B k = ^? A'jB], 



/ON 

(33) 

j = 1 

Furthermore the derivatives of a scalar, 

(34) B, = g (,' = 1, 2, 3, 4) 

transform like the components of a four-vector and hence the four- 
dimensional Laplacian of an invariant scalar, 



is also invariant to an orthogonal transformation. 

1.20. Elements of Tensor Analysis. Although most physical quanti- 
ties may be classified either as scalars, having only magnitude, or as 
vectors, characterized by magnitude and direction, there are certain 
entities which cannot be properly represented by either of these terms. 
The displacement of the center of gravity of a metal rod, for example, 
may be defined by a vector; but the rod may also be stretched along the 
axis by application of a tension at the two ends without displacing the 
center at all. The quantity employed to represent this stretching must 
thus indicate a double direction. The inadequacy of the vector concept 
becomes all the more apparent when one attempts the description of a 
volume deformation, taking into account the lateral contraction of the 



SBC. 1.20] ELEMENTS OF TENSOR ANALYSIS 65 

rod. In the present section we shall deal only with the simpler aspects 
of tensor calculus, which is the appropriate tool for the treatment of such 
problems. 

In a three-dimensional continuum let each rectangular component 
of a vector B be a linear function of the components of a vector A. 

Bi = TuAi + T 12 A Z + T 13 A 3 , 

(36) B, = rAi + T 22 A 2 + T 28 A 8 , 

3 = T 9l Ai + T 82 A 2 + 



In order that this association of the components of B with the components 
of A in the system X be preserved as the coordinates are rotated, it is 
necessary that the coefficients T& transform in a specific manner. The 
Tjk are therefore variant scalar s. A tensor or more properly, a tensor of 
rank two will now be defined as a linear transformation of the com- 
ponents of a vector A into the components of a vector B which is invariant 
to rotations of the coordinate system. The nine coefficients Tjk of the 
linear transformation are called the tensor components. 

To determine the manner in which a tensor component must trans- 
form we write first (36) in the abbreviated form 

(37) B t = 2 T, k A k , (j = 1, 2, 3). 

fc = l 

If (37) is to be invariant to the transformation defined by 

3 3 

(38) x'j = V ajkx k , V a 3l a ]k = d lk , 

* - 1 j-i 

then the Tj k must transform to T' {1 such that 

(39) ff t = V TV',, (,' = 1, 2, 3). 



Multiply (37) by a/ and sum over the index j. 

q q o 

(40) J a^Bf = V V a*5P*A*. 
>-i j-i*-i 

But 

(41) B( = 2 a*B,, A k = V att AJ, 

= i zT\ 

and, hence, 

333 1 

(42) 3 = 2 (S S a '*^*) ^ = X n^',. 

j-i v ytit-i ' ffi 



66 THE FIELD EQUATIONS [CHAP. I 

The components of a tensor of rank two transform according to the law 

3 3 

(43) r - a a " T *> ft i - 1, 2, 3); 



inversely, any set of nine quantities which transform according to (43) 
constitutes a tensor. 

By an analogous procedure one can show that the reciprocal trans- 
formation is 



If the order of the indices in all the components of a tensor may be 
changed with no resulting change in the tensor itself, so that Tjk = T k j, 
the tensor is said to be completely symmetric. A tensor is completely 
antisymmetric if an interchange of the indices in each component results 
in a change in sign of the tensor. The diagonal terms T }J - of an anti- 
symmetric tensor evidently vanish, while for the off-diagonal terms, 
T ik = - TV It is clear from (43) that if T ik = T ki , then also T^ = T' K . 
Likewise if T ik = T k] , it follows that T' it = T' K . The symmetric or 
antisymmetric character of a tensor is invariant to a rotation of the 
coordinate system. 

The sum or difference of two tensors is constructed from the sums or 
differences of their corresponding components. If 2 R is the sum of the 
tensors 2 S and 2 TY its components are by definition 

(45) R jk = S ]k + T ik , 0', k = 1, 2, 3). 

In virtue of the linear character of (43) the quantities Rj k transform like 
the Sj k and Tj k and, therefore, constitute the components of a tensor 2 R. 
From this rule it follows that any asymmetric tensor may be represented 
as the sum of a symmetric and an antisymmetric tensor. Assuming 2 R 
to be the given asymmetric tensor, we construct a symmetric tensor 2 S 
from the components 



(46) Sj k 

and an antisymmetric tensor 2 T from the components 

(47) T0 



Then by (45) the sum of 2 S and 2 T so constructed is equal to 2 R. 

In a three-dimensional manifold an antisymmetric tensor reduces to 
three independent components and in this sense resembles a vector. The 

1 Tensors of second rank will be indicated by a superscript as shown. 



SEC. L20] ELEMENTS OF TENSOR ANALYSIS 67 

tensor (36), for example, reduces in this case to 

B! = - T n A 2 + T 13 A 3 , 

(48) J9 2 = T^Ai + - r. 8 A,, 
3 - -Ti 3 Ai + T 32 A 2 + 0. 

These, however, are the components of a vector, 

(49) B = T X A, 
wherein the vector T has the components 

(50) T, = 7*32, T 2 = r w , Z r , = Z'n. 

Now it will be recalled that in vector analysis it is customary to distin- 
guish polar vectors, such as are employed to represent translations and 
mechanical forces, from axial vectors with which there are associated 
directions of rotation. Geometrically, a polar vector is represented by a 
displacement or line, whereas an axial vector corresponds to an area. A 
typical axial vector is that which results from the vector or cross product 
of two polar vectors, and we must conclude from the above that an axial 
vector is in fact an antisymmetric tensor and its components should 
properly be denoted by two indices rather than one. Thus for the com> 
ponents of T = A X B we write 

(51) Tik = A iB k - AtB, = -ZV, (j, fc = 1, 2, 3). 

If the coordinate system is rotated, the components of A and B are 
transformed according to 

(52) 



= V anAJ, B k = V a ik B[. 
i-i i-a 



Upon introducing these values into (51) we find 

3 3 

(53) 

a relation which is identical with (44) and which demonstrates that the 
components of a "vector product" of two vectors transform like the 
components of a tensor. The essential differences in the properties of 
polar vectors and the properties of those axial vectors by means of which 
one represents angular velocities, moments, and the like, are now clear: 
axial vectors are vectors only in their manner of composition, not in 
their law of transformation. It is important to add that an antisym- 
metric tensor can be represented by an axial or pseudo-vector only in a 
three-dimensional space, and then only in rectangular components. 



68 THE FIELD EQUATIONS [CHAP. I 

Since the cross product of two vectors is in fact an antisymmetric 
tensor, one should anticipate that the same is true of the curl of a vector. 
That the quantity dAi/dx*, where A, is a component of a vector A, is 
the component of a tensor is at once evident from Eq. (22). 



The components of v X A, 



therefore, transform like the components of an antisymmetric tensor. 
The divergence of a tensor is defined as the operation 

(56) (div T), = 



The quantities Bj are easily shown to transform like the components of 
a vector. 



or, on summing over I and applying the conditions of orthogonality, 



- 



< -23? -;- 2? *> 

Isl j 08 ! Af^l J 1 






divergence of a tensor of second rank is a vector, or tensor of first rank. 
The divergence of a vector is an invariant scalar, or tensor of zero rank. 
These are examples of a process known in tensor analysis as contraction. 
As in the case of vectors, the tensor concept may be extended to mani- 
folds of four dimensions. Any set of 16 quantities which transform 
according to the law, 



(59) n = a * r *' (i ' l = *' 2 ' 3 ' 

;-l ffl 

or its reciprocal, 

4 4 

(60) T, 



will be called a tensor of second rank in a four-dimensional manifold. 
As in the three-dimensional case, the tensor is said to be completely 



SBC. 1.21] THE SPACE-TIME SYMMETRY 69 

symmetric if T )k = T ki , and completely antisymmetric if !> = T kit 
with TJJ = 0. In virtue of its definition it is evident that an anti- 
symmetric four-tensor contains only six independent components. 
Upon expanding (59) and replacing T ki by - T jk , then re-collecting terms, 
we obtain as the transformation formula of an antisymmetric tensor the 
relation 

_i_-JL 44 

(61) 




Any six quantities that transform according to this rule constitute an 
antisymmetric four-tensor or, as it is frequently called, a six-vector. 

In three-space the vector product is represented geometrically by the 
area of a parallelogram whose sides are defined by two vectors drawn 
from a common origin. The components of this product are then the 
projections of the area on the three coordinate planes. By analogy, the 
vector product in foUr-space is defined as the "area" of a parallelogram 
formed by two four-vectors, A and B, drawn from a common origin. 
The components of this extended product are now the projections of the 
parallelogram on the six coordinate planes, whose areas are 

(62) T jk = AtB k - A k B f = -T fc/J (j, k = 1, 2, 3, 4). 

The vector product of two four-vectors is therefore an antisymmetric 
four-tensor, or six-vector. 

If again A is a four-vector, the quantities 

(63) T '* = ^-!=- T *>> , * - 1, 2, 8, 4), 

can be shown as in (54) to transform like the components of an anti- 
symmetric tensor. The T jk may be interpreted as the components of 
the curl of a four-vector. 

As in the three-dimensional case the divergence of a four-tensor is 
defined by 



<*> (div2T) < = ' 0' = 1,2, 3, 4), 



a set of quantities which are evidently the components of a four-vector. 
1.21. The Space-time Symmetry of the Field Equations. A remark- 
able symmetry of form is apparent in the equations of the electromagnetic 
field when one introduces as independent variables the four lengths 

(65) xi = Xj Xz = y, #3 = 3, Xt = ict, 

where c is the velocity of light in free space. When expanded in rec- 



70 THE FIELD EQUATIONS [CHAP. I 

tangular coordinates, the equations 

(II) V X H - $9 = J, (IV) V D = p, 
are represented by the system 

n TT ft TT *\ T\ 

___ _ I C\ I _, ~ __ a ft . T 

__ j yj _j_ ___ ^ ^_ __ ^ g 

, AA x dxi dxs dx^ 

(66) ATT ^ n 

d/i2 a/zi . A . djL/3 r 

. . -J- () . = J g^ 

ic -j~ 2*c ~j- ic -f- = icp. 

dxi dx% dx$ 

We shall treat the right-hand members of this system as the componenta 
of a "four-current" density, 

( fi.*7\ T T T - T T T T ' 

and introduce in the left-hand members a set of dependent variables 
defined by 

On = (?12 = HZ Gl3 = /: 

c* v f^ c\ r< u 

fW\ Zl = "~-" 3 ^2 = U (^23 = Hi 

(68) r T, r _ TT r<-n n ; 

Crsi = 112 ^32 -" 1 vJ"33 v VJT34 ^ 

Then in the reference system X y Eqs. (II) and (IV) reduce to 

4 .^ 

(69) 



Only six of the (?# are independent, and the resemblance of this set 
of quantities to the components of an antisymmetric four-tensor is 
obvious. Since the divergence of a four-tensor is a four-vector, it follows 
from (69) that if the G 3 k constitute a tensor, then the Jk are the com- 
ponents of a four-vector; inversely, if we can show that J is indeed a 
four-vector, we may then infer the tensor character of 2 G. However, 
we have as yet offered no evidence to justify such an assumption. In 
the preceding sections it was shown that the vector or tensor properties 
of sets of scalar quantities are determined by the manner in which they 
transform on passing from one reference system to another. Evidently 
an orthogonal transformation of the coordinates Xk corresponds to a 
simultaneous change in both the space coordinates x, y, z and the time t, 
and only recourse to experiment will tell us how the field intensities may 



SEC. 1.21] THE SPACE-TIME SYMMETRY 71 

be expected to transform under such circumstances. In Sec. 1.22 we 
shall set forth briefly the experimental facts which lead one to conclude 
that the Jk are components of a four-vector, and the Gjk the components 
of a field tensor; in the interim we regard (69) and the deductions that 
follow below merely as concise and symmetrical expressions of the field 
equations in a fixed system of coordinates. 
The two homogeneous equations 

\TQ 

(I) V X E + ~ = 0, (III) V B = 0, 
are represented by the system 



(70) 

3^2 _ aS 

ftci das, 



_J - *h _ *5 4. n _ n 
dxi dx 2 ' 



After division of the first three of these equations by ic, an antisym- 
metrical array of components is defined as follows: 

Fn = 

T) 

21 >3 

(71) 

T) 

d l J 2 



= B 3 


F, _ D 
13 #2 


77? ' 






C 


= 


Til ___ D 


P = _^ 






C 


= -Bi 


iy rv 


F 34 = -" 






c 


s> 


F 43 = ~ EZ 


^44 = 0. 



Then all the equations of (70) are contained in the system 

(72) ^ + ^JS + ^J* = o 

v ax fc ^ dx/ ^ dx U; 

where t, j, ft are any three of the four numbers 1, 2, 3, 4. 

The arrays (68) and (71) are congruent in the sense that in each the 
real components pertain to the magnetic field, while the imaginary com- 
ponents are associated with the electric field. To indicate this partition 
it is convenient to represent the sets of components by the symbols 



(73) P = (B, - j E\ ^G = (H, -tcD). 

\ c / 



72 THE FIELD EQUATIONS [CHAP. I 

Now the field equations may be defined equally well in terms of the 
"dual" systems: 



(74) 


*F* = I _f E, Bj, 


2 G*= (-tcD,H), 




or 








/ 


? S = FA = -- E, 


Fft = - E% F 


1*4 = Bl 




^ c 


c 




/ 


7* = 1 j, F ? s = 


< 


'& = -82 


(75) 


C 


23 "" c x 




F 


'* * P 1 p* * ' p 
si ~ - ^2 r 32 = - 11 


TTT^ _ f\ Jjl 


3*4 = -B, 




C C 






t 


i-fj = _5j f = -B 2 


F# D T?> 
AS == ~"~ f> i 


4*4 = 0, 



and a corresponding system for the components of 2 G*. Upon intro- 
ducing these values into (I) and (III), and (II) and (IV), respectively, 
we obtain 

(76) 2l3 = ' (j= 1,2, 3, 4), 



+ l + r = J '' ft* M - 1. * 3, 4). 

It has been pointed out by various writers that this last representation 
is artificial, in that (74) implies that E is an axial vector in three-space 
and B a polar vector, whereas the contrary is known to be true. The 
representation 



<"> + + - ' .*- !,*!,>. 

(69) ' J " 0-1,2,3,4), 



must in this sense be considered the " natural" form of the field equations. 
To these we add the equation of continuity, 

(V) ^J + ^ = ' 

which in four-dimensional notation becomes 

(78) 

If the components Fjk are defined in terms of the components of a 
"four-potential" * by 

(79) F,, - - g, O; * - 1, 2, 3, 4), 



SBC. 1.21] THE SPACE-TIME SYMMETRY 73 

one may readily verify that Eq. (72) is satisfied identically. Now in 
three-space the vectors E and B are derived from a vector and scalar 
potential. 

(80) E = -V* - ^, B = v x A; 

or, in component form, 

dA >' dAk dA >' 

- ; >i = - -- - 

*4 dx/ da:*' 

where the indices t, j, fc are to be taken in cyclical order. Clearly all 
these equations are comprised in the system (79) if we define the com- 
ponents of the four-potential by 

(82) *! = A x , $ 2 = A 9 , *, = A,, * 4 = - *. 

c 

As in three dimensions, the four-potential is useful only if we can 

determine from it the field 2 G(H, -tcD) as well as the field 2 F(B, -~ E). 

c 

Some supplementary condition must, therefore, be imposed upon 4> in 
order that it satisfy (69) as well as (72); thus it is necessary that the 
components G jk be related functionally to the Fj k . We shall confine the 
discussion here to the usual case of a homogeneous, isotropic medium and 
assume the relations to be linear. To preserve symmetry of notation 
it will be convenient to write the proportionality factor which charac- 
terizes the medium as y/*, so that 

(83) Q ih = 7)k Fj k ; 
but it is clear from (68) and (71) that 1 

(84) y ik = - when j, fc = 1, 2, 3, y ik = cc 2 when j or k == 4. 

These coefficients are in fact components of a symmetrical tensor, and 
with a view to subsequent needs the diagonal terms are given the values 

(85) -YH = 1, (j = 1, 2, 3), T44 = /ic 2 c. 
Equation (69) IB now to be replaced by 

(86) * = J >> 0'= 1,2, 3, 4). 



1 A medium which is anisotropic in either its electrical properties or its magnetic 
properties may be represented as in (83) provided the coordinate system is chosen to 
coincide with the principal axes. This also is the case if its principal axes of elec- 
trical anisotropy coincide with those of magnetic anisotropy. 



74 THE FIELD EQUATIONS [CHAP. I 

Upon introducing (79) we find that (86) is satisfied, provided O is a 
solution of 

4 

(87) y, ~ (7**,-) = - Ji, (j = 1, 2, 3, 4), 



^ 

subject to the condition 

4 

(88) 



This last relation is evidently equivalent to 

(89) V.A + M 6^ = 0, 
and (87) comprises the two equations 

V*A f - ^L' = -fj it (j = 1, 2, 3), 

(90) vV _ Me ^=_i p . 

In free space /io*o = c~ 2 , 7,-* = /xo" 1 , for all values of the indices. Equa- 
tions (87) and (88) then reduce to the simple form: 



= 1 A; = 1 

1.22. The Lorentz Transformation. The physical significance of 
these results is of vastly greater importance than their purely formal 
elegance. A series of experiments, the most decisive being the celebrated 
investigation of Michelson and Morley, 1 have led to the establishment of 
two fundamental postulates as highly probable, if not absolutely certain. 
According to the first of these, called the relativity postulate, it is impossible 
to detect by means of physical measurements made within a reference 
system X a uniform translation relative to a second system X'. That 
the earth is moving in an orbit about the sun we know from observations 
on distant stars; but if the earth were enveloped in clouds, no measure- 
ment on its surface would disclose a uniform translational motion in 
space. The course of natural phenomena must therefore be unaffected 
by a nonaccelerated motion of the coordinate systems to which they are 
referred, and all reference systems moving linearly and uniformly relative 
to each other are equivalent. For our present needs we shall state the 
relativity postulate as follows: When properly formulated , the laws of 

1 MICHELSON and MORLBT, Am. J. Sd., 3, 34, 1887. 



SBC. 1.221 THE LORENTZ TRANSFORMATION 75 

physics are invariant to a transformation from one reference system to 
another moving with a linear , uniform relative velocity. A direct conse- 
quence of this postulate is that the components of all vectors or tensors 
entering into an equation must transform in the same way, or covariantly. 
The existence of such a principle restricted to uniform translations was 
established for classical mechanics by Newton, but we are indebted to 
Einstein for its extension to electrodynamics. 

The second postulate of Einstein is more remarkable: The velocity 
of propagation of an electromagnetic disturbance in free space is a universal 
constant c which is independent of the reference system. This proposition 
is evidently quite contrary to our experience with mechanical or acoustical 
waves in a material medium, where the velocity is known to depend 
on the relative motion of medium and observer. Many attempts have 
been made to interpret the experimental evidence without recourse to 
this radical assumption, the most noteworthy being the electrodynamic 
theory of Ritz. 1 The results of all these labors indicate that although a 
constant velocity of light is not necessary to account for the negative 
results of the Michelson-Morley experiment, this postulate alone is con- 
sistent with that experiment and other optical phenomena. 2 

Let us suppose, then, that a source of light is fixed at the origin 
of a system of coordinates X(x, y, z). At the instant t = 0, a spherical 
wave is emitted from this source. An observer located at the point 
x, y,zinX will first note the passage of the wave at the instant ct, and the 
equation of a point on the wave front is therefore 

(92) x 2 + 7/ 2 + z 2 - cH 2 = 0. 

The observer, however, is free to measure position and time with respect 
to a second reference frame X r (x' y y' , z') which is moving along a straight 
line with a uniform velocity relative to O. For simplicity we shall assume 
the origin O' to coincide with O at the instant t = 0. According to the 
second postulate the light wave is propagated in X' with the same 
velocity as in X t and the equation of the wave front in X r is 

(93) x'" + y 1 ' + z 1 ' - cV* = 0. 

By t' we must understand the time as measured by an observer in X' 
with instruments located in that system. Here, then, is the key to the 
transformation that connects the coordinates #, t/, z, t of an observation 
or event in X with the coordinates x' 9 y' t z', t' of the same event in X 1 : it 
must be linear and must leave the quadratic form (92) invariant. The 
linearity follows from the requirement that a uniform, linear motion of a 
particle in X should also be linear in X f . 

1 RITZ, Ann. chim. phys., 13, 145, 1908. 

1 An account of these investigations will be found in Pauli's article, loc. cit. t p. 549. 



76 



THE FIELD EQUATIONS 



[CHAP. I 



Let 

(94) xi = x, x z = y, x* = z, X* = tc*, 

be the components of a vector R in a four-dimensional manifold 

X(Xi, X*, X 3 , 2 4 ). 

(95) fi 2 = sj + x\ + x\ + x\. 

The postulate on the constancy of the velocity c will be satisfied by the 
group of transformations which leaves this length invariant. But in 
Sec. 1.19 it was shown that (95) is invariant to the group of rotations in 
four-space and we conclude, therefore, that the transformations which 
take one from the coordinates of an event in X to the coordinates of that 
event in X' are of the form 

(26) *J = 2 a,**, (j = 1, 2, 3, 4), 



where 
(27) 



k = X > 2 > 3 > 



the determinant |a/jfc| being equal to unity. 

We have now to find these coefficients. The calculation will be 
simplified if we assume that the rotation involves only the axes # 3 and X*, 
and the resultant lack of generality is inconsequential. We take, there- 
fore, x'i = Xij x' 2 = x 2 , and write down the coefficient matrix as follows: 



(96) 



The conditions of orthogonality reduce to 

(97) 033 -J- <1 4 3 = 1, #34 -f" & 44 = 1, &33&34 T" &43& 4 4 == 0. 

If we put a 33 = a, #34 = ictp, we find from (97) that a 44 = a, 
43 = ^f-iaP, <x -\/l /8 2 = 1. Only the upper sign is consistent with 
the requirement that the determinant of the coefficients be positive unity, 
and this in turn is the necessary condition that the group shall contain 
the identical transformation. In terms of the single parameter ft the 
coefficients are 

(98) 1X33 =5: #44 =" - '*-- " - 





Xl 


*2 


*3 


0?4 


si 


1 











x t 





1 








4 








fl53 


a 34 


*; 








&4S 


<Z 4 4 



SEC. 1.22] THE LORENTZ TRANSFORMATION 77 

for the transformation itself, we have 

(99) x( = xi, z' 2 = 3 2 , 



Reverting to the original space-time manifold this is equivalent to 
x' = x, y f = t/, 

(ioo) w- 



The parameter ft may be determined by considering x', y', z' to be the 
coordinates of a fixed point in X f . The coordinates of this point with 
respect to X are #, y, z. Since dz f = 0, it follows that 



and hence the rotation defined in (96) and (97) is equivalent to a transla- 
tion of the system X' along the z-axis with the constant velocity v relative 
to the unprimed system X. 
The transformation 



(102) 




y' 



obtained from (100) by substitution of the value for ft, or its inversion, 
(103) z = -7=L= (' + t>O, = 



has been named for Lorentz, who was the first to show that Maxwell's 
equations are invariant with respect to the change of variables defined 
by (102), but not invariant under the "Galilean transformation : " 

(104) z' = z - vt, t r = t. 

All known electromagnetic phenomena may be properly accounted for 
if the position and time coordinates of an event in a moving system X' 
be related to the coordinates of that event in an arbitrarily fixed system 
X by a Lorentz transformation. The Galilean transformation of classical 
mechanics represents the limit approached by (102) when v c, and 
may be interpreted as the relativity principle appropriate to a world in 
which electromagnetic forces are propagated with infinite velocity. 



78 THE FIELD EQUATIONS [CHAP. I 

1.23. Transformation of the Field Vectors to Moving Systems. We 

shall not dwell upon the manifold consequences of the Lorentz transforma- 
tion; the Fitzgerald-Lorentz contraction, the modified concept of simul- 
taneity, the variation in apparent mass, the upper limit c which is imposed 
upon the velocity of matter, belong properly to the theory of relativity. 
The application of the principles of relativity to the equations of the 
electromagnetic field is essential, however, to an understanding of the 
four-dimensional formulation of Sec. 1.21. 

The Lorentz transformation has been deduced from the postulate on 
the constancy of the velocity of light and has been shown to be equivalent 
to a rotation in a space xi, rc 2 , 3 , 4 = ict. Now according to the rela- 
tivity postulate, the laws of physics, when properly stated, must have 
the same form in all systems moving with a relative, uniform motion; 
otherwise, it would obviously be possible to detect such a motion. In 
Sees. 1.19 and 1.20 it was shown that the curl, divergence, and Laplacian 
of vectors and tensors in a four-dimensional manifold are invariant to a 
rotation of the coordinate system. Therefore, to ensure the invariance 
of the field equations under a Lorentz transformation it is only necessary 
to assume that the four-current J and the four-potential 4> do indeed 
transform like vectors, and that the quantities 2 F, 2 G transform like 
tensors. In other words, we base the vector and tensor character of these 
four-dimensional quantities directly on the two postulates. 

The four-current J satisfies the equation 



Under a rotation of the coordinate system the components transform as 

4 

(105) J} 



or, upon introducing the values for a ik from (98), 

J 1 T 
<Jx Jx, 

1 



with its inverse transformation 

J - = 



We shall assume henceforth that the reference system X f is fixed within a 
material body which moves with the constant velocity v relative to the 



SEC. 1.23] TRANSFORMATION OF THE FIELD VECTORS 79 

system X. This latter may usually be assumed at rest with respect to 
the earth. If the velocity v is very much less than the speed of light, 
Eq. (107) is approximately equal to 

(108) J, = JJ + vp', P = P '. 

An observer on the moving body measures a charge density p' and a 
current density J' z ] his colleague at rest in X finds the current J' z aug- 
mented by the convection current vp'. 

In like manner the relations between the electric and magnetic 
vectors defining a given field in a fixed and in a moving system are 
obtained directly from the rule (61) for the transformation of the com- 
ponents of an antisymmetric tensor. Upon substitution of the appro- 
priate values for the coefficients a#, one obtains for the components of 2 F : 



iz + 

F u = 



W #22^33^23 + #22^34^24 = 



24 = 




and, hence, 




The restriction to translations along the z-axis may be discarded by 
writing v as a vector representing the translational velocity of X' (the 
moving body) in any direction with respect to a fixed system X. Since in 
(110) the orientation of the z-axis was arbitrary, we have in general 



(Ill) *'-*' * 

5'j. = _ 2 f B - - t v X E J , 



80 THE FIELD EQUATIONS [CHAP. I 

where || denotes components parallel, J, components perpendicular to 
the axis of translation. Dropping terms in c~ 2 , as is justifiable whenever 
the body is moving with a velocity v 3 X 10 8 meters/second, we 
obtain the approximate formulas 



L, E' = 

The implication of these results is striking indeed: the electric and 
magnetic fields E and B have no independent existence as separate 

__ 7* 

entities. The fundamental complex is the field tensor 2 F = (B, E) ; 

c 

the resolution into electric and magnetic components is wholly relative 
to the motion of the observer. When at rest with respect to permanent 
magnets or stationary currents, one measures a purely magnetic field B. 
An observer within a moving body or system X f , on the other hand, 
notes approximately the same magnetic field, but in addition an electro- 
static field of intensity E' = v X B. Or, inversely, the moving body 
may carry a fixed charge. To an observer on the body, moving with the 
charge, the field is purely electrostatic, whereas his colleague aground 
finds a magnetic field in company with the electric, identifying quite 
rightly the moving charge with a current. 

From the tensor 2 G = (H, icD) are calculated in like fashion the 
transformations of the vectors H and D from a fixed to a moving system. 

(113) **- H 

(H - v X 




The invariance of Maxwell's equations to uniform translations amounts 
to this: if the vector functions E, B, H, D define an electromagnetic field 
in a system X, the equations 

/)TV 
Y'XE' + 2g- = 0, V'-B' = 0, 

(114) * 

V X H' - - = J', V'-D' = p', 

are satisfied in a system X r which moves with the constant velocity v 
relative to X, the operator V' implying that differentiation is to be effected 
with respect to the variables x', y', z'. An observer at rest in X' inter- 
prets the vectors E', B', H', D' as the intensities of an electromagnetic 
field satisfying Maxwell's equations. Clearly the ratios of D to E and 
H to B are not preserved in both systems. The macroscopic parameters 



SEC. 1.231 TRANSFORMATION OF THE FIELD VECTORS 81 

, jit, <r are also subject to transformation, which may be ascribed to an 
actual change in the structure of matter in motion. In practice one is 
interested usually in the mechanical and electromotive forces, measured in 
the fixed system X, which act on moving matter , rather than in the trans- 
formed field intensities E', B', IF, D'. The determination of these 
forces and of the differential equations which they satisfy within the 
framework of the relativity theory was accomplished by Minkowski in 
the course of his investigation on the electrodynamics of moving bodies. 
The vector character of the four-potential is demonstrated by Eq. 
(79) which expresses the field tensor 2 F as the curl of 4>. Under a Lorentz 
transformation 



(115) *} - 

or, in terms of vector and scalar potentials, 




As in the case of the field vectors, the resolution into vector and scalar 
potentials in three-space is determined by the relative motion of the 
observer. 

In conclusion it may be remarked that a rotation of the coordinate 
system leaves invariant the scalar product of any two vectors. It was 
in fact from the required invariance of the quantity 

(117) R R = R* = x* + y* + z 2 - c 2 * 2 , 

that we deduced the Lorentz transformation. Since the current density 
J and the potential I> have been shown to be four-vectors, it follows that 
the quantities 

J 2 = Jl + Jl + Jl - cV, 

(118) $ 2 = Al + Al + Al - , 

c 

^ J = AJ X + AyJy + AJ, fa, 

are true scalar invariants in a space-time continuum. There are, more- 
over, certain other scalar invariants of fundamental importance to the 
general theory of the electromagnetic field. From the transformation 
formula Eq. (59) the reader will verify that if 2 S and *T are two tensors 
of second rank, the sums 



kj = inyariant > 

y-i *-i 



82 THE FIELD EQUATIONS [CHAP. I 

are invariant to a rotation of the coordinate axes. These quantities 
may be interpreted as scalar products of the two tensors. Let us form 
first the scalar product of 2 F with itself. According to (71) and (119) 
we find 

44 / 1 \ 

(120) F * = 2 B * " 2 E = invariant ' 



Next we construct the scalar product of 2 F with its dual 2 F* defined in 
(75). 

4 4 

(121) ^ ^^ FjkFJk 4 - B E = invariant. 

From the tensor 2 G and its dual 2 G* may be constructed the invariants 

4 4 

(122) "V^ ^? Oj k = 2(H 2 c 2 Z) 2 ) = invariant, 



. 1 

4 4 



(123) ^ (?y * (?J * ^ " 4lcH * D = invariant - 

>-! A;l 

Proceeding in the same fashion, we obtain 



= invariant, 
I D H E H ) 

= invariant. 

The invariance of these quantities in configuration space is trivial; they 
are set apart from other scalar products by the fact that they preserve the 
same value in every system moving with a uniform relative velocity. 



CHAPTER II 
STRESS AND ENERGY 

To translate the mathematical structure developed in the preceding 
pages into experiments which can be conducted in the laboratory, we 
must calculate the mechanical forces exerted in the field upon elements of 
charge and current or upon bodies of neutral matter. In the present 
chapter it will be shown how by an appropriate definition of the vectors 
E and B these forces may be deduced directly from the Maxwell equations. 
In the course of this investigation we shall have to take account of the 
elastic properties of material media. A brief digression on the analysis of 
elastic stress and strain will provide an adequate basis for the treatment 
of the body and surface forces exerted by electric or magnetic fields. 

STRESS AND STRAIN IN ELASTIC MEDIA 

2.1. The Elastic Stress Tensor. Let us suppose that a given solid or 
fluid body of matter is in static equilibrium under the action of a specified 
system of applied forces. Within 
this body we isolate a finite volume 
V by means of a closed surface S, as 
indicated in Fig. 12. 

Since equilibrium has been as- 
sumed for the body and all its parts, 
the resultant force F exerted on the 
matter within S must be zero. Con- 
tributing to this resultant are volume 
or body forces, such as gravity, and 
surface forces exerted by elements of 
matter just outside the enclosed 
region on contiguous elements 

within. Throughout 7, therefore, FK- 12. A region V bounded by a surface 
. . , ,. , ., , j S in an elastic medium under strow. 

we suppose force to be distributed 

with a density f per unit volume, while the force exerted by matter outside 
S on a unit area of S will be represented by the vector t. The components 
of t are evidently normal pressures or tensions and tangential shears. The 
condition of translational equilibrium is expressed by the equation 

(1) 

V ' 

83 




84 STRESS AND ENERGY [CHAP. II 

To ensure rotational equilibrium it is necessary aluo that the resultant 
torque be zero, or 

(2) 



where r is the radius vector from an arbitrary origin to an element of 
volume or surface. 

The transition from the integral relations (1) and (2) to an equivalent 
differential or local system expressing these same conditions in the 
immediate neighborhood of an arbitrary point P(z, y, z) may be accom- 
plished by a method employed in Chap. I. The surface S is shrunk 
about P and the components of t are expanded in Taylor series. The 
integral can then be evaluated over the surface and on passing to the 
limit the conditions of equilibrium are obtained in terms of the deriva- 
tives of t at P. This labor may be avoided by applying theorems already 
at our disposal. Let n be the unit, outward normal to an element of S. 
There are then three vectors X, Y, Z which satisfy the equations 

(3) k = X n, t v = Y n, t, = Z n. 

The quantity X n is clearly the x-component of force acting outward on a 
unit element of area whose orientation is fixed by the normal n. The 
expansion of the scalar products in the form 

t x = X X U X + XyUy + X Z U Zy 

(4) t v = Y x n x + YyUy + Y z n zy 
t z = Z x n x + ZyUy + Z z n t 

may also be interpreted as a linear transformation of the components of 
n into the components of t, the components n x , n yj n z being the direction 
cosines of n with respect to the coordinate axes. The equilibrium of the 
rr-components of forces acting on matter within S is now expressed by 



(5) 

which in virtue of the divergence theorem and the arbitrariness of V is 
equivalent to the condition that at all points within S 

(6) f x + V X = 0. 
For the y- and ^-components we have, likewise, 

(7) f y + V Y = 0, /. + V Z = 0. 

The rotational equilibrium expressed by (2) imposes further condi- 
tions upon the nine components of the three vectors X, Y, Z. The 
^-component of this equation is, for example, 

(8) f y (yf* - */) dv + f a (yt. - zt y ) da = 0. 



SBC. 2.1] THE ELASTIC STRESS TENSOR 85 

Introduction of the values of t, and < defined in (3) leads to 

(9) f r (yf. - zf v ) dv + f s (yZ - Y) n da = 0, 

which, again thanks to the divergence theorem and the arbitrariness of 
V, is equivalent to 

(10) yf, -tf, + V> (yZ - Y) = 0. 
But 

(11) V-foZ) = 2/V-Z + Z- Vy - yV-Z + Z tt . 
Equation (10) reduces to 

(12) y(f. + V . Z) - *(/ + v . Y) + Z v - Y. = 0; 
or, on taking account of (7), 

(13) Z v = Y,. 

In like manner there may be derived from the y- and ^-components of 
(2) the symmetry relations 

(14) X y = F., X z = Z,. 

The nine components X x , X v , . . . Z t , representing forces exerted on 
unit elements of area, are called stresses. The diagonal terms X x , Y v , Z z 
act in a direction normal to the surface element and are, therefore, 
pressures or tensions. The remaining six components are shearing 
stresses acting in the plane of the element. These nine quantities con- 
stitute the components of a symmetrical tensor, as is evident from (4) 
and the fact that t and n are true vectors (cf. Sec. 1.20). For the sake of 
a condensed notation we shall henceforth represent the components of the 
stress tensor by T,-*, where 

(15) X x = Tn, X y = T 12 , ... Y, = T M , ... Z, = r 3a . 

In order that a fluid or solid medium under stress shall be in static 
equilibrium it is necessary that at every point 

(16) f + div 2 T = 0, Tik = 2V 

Imagine an infinitesimal plane element of area containing a point 
(x, y, z) in a stressed medium. The stresses acting across this surface 
element will, in general, be both normal and tangential, but there are 
three distinct orientations with respect to which the stress is purely 
normal. Now if the resultant force t acting on unit area of a plane ele- 
ment is in the direction of the positive normal n, one may put 

(17) t - Xn, 



86 STRESS AND ENERGY [CHAP. II 

where X is an unknown scalar function of the coordinates x, y, z, of the 
element. When this condition is imposed upon Eqs. (4) one obtains the 
homogeneous system 

(18) Xn, = V T jk n k , (j = 1, 2, 3). 



In order that these homogeneous equations be self-consistent, it is 
necessary that their determinant shall vanish; whence it follows that the 
scalar function X can be determined from 



(19) 



X 



7*22 "~ X TW 

JL32 TzZ X 



= 0, 



provided, of course, that the stress components T ]k with respect to some 
arbitrary coordinate system are known. The secular equation (19) has 
three roots, X a , X&, X c , and these fix, through Eq. (18), three orientations of 
the surface element which we shall designate respectively as n (a) , n (6) , n (c) . 
If the roots are distinct, the preferred directions defined by the unit 
vectors n (o) , n (6) , n (c) called the principal axes of stress are mutually 
perpendicular. Let us consider, for example, n (a) and n< 6) . According 
to (18), 

(20) X a rc<.*> = 



Multiply the first of these equations by nf\ the second by n< 0) , subtract 
the second from the first and sum over j. 



(21) (Xa - X 6 ) V n 

~l ,--!*-! 

The right-hand sum vanishes, leaving 

(22) (X a - X 6 )n(> - n< = 0. 

If X a 5^ X&, the vector n (o) must be orthogonal to n (6) as stated. 

The physical significance of the principal axes may be made clear in 
another manner. At a point P in a stressed medium the symmetrical 
tensor 2 T associates with a unit vector n the resultant force t acting on a 
unit surface element normal to n. Let the origin of a rectangular coordi- 
nate system be located at P. The scale of length in this new system is 
arbitrary and we may suppose that the components of n, drawn from P, 
are , 77, f . Now the scalar product of the vectors t and n, which we shall 
call <, is a quadratic in , ?y, f . 

(23) *(t 77, f ) = t n = 



SEC, 2.2] 



ANALYSIS OF STRAIN 



87 



The surface < =* constant is called the stress quadric. By a rotation of 
the coordinate axes (23) can be reduced to the square terms alone, and 
it is clear that the principal axes n (o) , n (6) , n (c) must coincide with the 
principal axes of the quadric. One will observe, furthermore, that 

(24) t = iV$. 

This property enables one to find the stress across any surface element at 
P by graphical construction. We shall suppose the stress to be such that 
the quadric is an ellipsoid and draw about P the surface $ = 1. The 
radius vector from P to any point P' on this surface is n. According to 
(24) the resultant force acting on an element at P whose normal is n is in 
the direction of V$, or normal to $ = 1 at P f . The magnitude of t 
according to (23) is equal to the reciprocal of the projection NP f indicated 




FIG. 13. Graphical determination of stress on an element of area at the point P from the 

stress quadric. 

in Fig. 13. Along the principal axes, a, fc, e, the normal to the surface 
$ = 1 coincides with the vector n and the stress on the element at P is 
purely normal. If these principal stresses are known, the quadric can 
be constructed and the stress on an arbitrarily oriented element deter- 
mined graphically. 

2.2. Analysis of Strain. If surface and volume forces arc applied to 
a perfectly rigid body, the resultant motion may be described in terms of 
a translation of the body as a whole and a rotation about its center of 
mass. If, however, the body is deformable, its parts suffer also relative 
displacements which increase until equilibrium is reestablished by internal 
forces evoked in the deformation. The changes in the relative positions 
of the parts of a body under stress are called strains. 

In an unstressed, continuous medium a point P is located with respect 
to an arbitrary, fixed origin by the radius vector r. Under the influence 
of applied forces the medium undergoes a deformation, in the course of 
which the matter located initially at P moves to a point P' at r' = r + s. 
The displacement s corresponding to a given deformation depends on 



88 STRESS AND ENERGY [CHAP. II 

the coordinates of the initial point P; we assume s to be a continuous 
function of r. Consider now any neighboring point Pi located by the 
radius vector ti. The initial position of PI with respect to P is fixed by 
the vector 5r, where 

(25) Sr = n - r. 

The displacement of PI during the deformation is Si, a function of ri. 
The relative displacement of two neighboring points P and PI occasioned 
by the deformation is therefore 

(26) 6s = Si s = s(r + 5r) s(r). 

Since s(r + 3r) is continuous, it may be expanded in a Taylor series; 
for points in a region sufficiently small about P we need retain only the 
linear term 

(27) 5s = (6r- V)s; 
or, in component form, 

(28) ta ' 



(The indices 1, 2, 3 again replace the subscripts x, y, z for convenience in 
summing.) 

The nine derivatives dsj/dxk are the components of a tensor (cf. 
Eq. (54), Sec. 1.20) which is in general asymmetrical. However, it can 
be resolved after the manner of Eqs. (46) and (47), page 66, into a 
symmetric part, 

(29) <"* = 

and an antisymmetric part, 



The components of this antisymmetric tensor are evidently identical with 
the components of an axial vector b, defined by 

(31) b = iV X s, 

(32) 6l = &S2, &2 = bl3, &3 = &2L 

The relative displacement 5s can likewise be split up into a part 5s' 
associated with the symmetric tensor a#, and a part 5s ;/ associated with 
the antisymmetric tensor b^. Then, by (28) and (30), 

(33) 5s" = b X 5r - i(V X s) X 8r. 

Physically this implies that the matter contained within an infinitesimal 
volume about P is subjected to a rotation as a rigid solid; the rotation 



Sac. 2.2] ANALYSIS OF STRAIN 89 

takes place about an instantaneous axis through P in the direction of 
V X s, and in circular measure is equal to i|V X s|. This local rotation 
is brought about by stresses whose curl does not vanish. 

Whereas the relative positions of points in the immediate vicinity 
of P are preserved by the local rotation (33), the symmetric tensor (29) 
defines a local deformation an actual stretching and twisting. We ask 
first whether there are any vectors 5r drawn from P whose direction is 
unchanged after the deformation; that is to say, are there any points 
PI, Fig. 14, which in the course of the 
deformation will move along the line 
defined by 5r? The necessary condi- 
tion is that 

(34) 5s' = X 5r, 

where X is an unknown scalar function 
of the coordinates of P; or 

3 

(35)X to, = a ik dx k , (j = 1, 2, 3). 

O 

The parameter X is found from the FlQ - 14. -Vectors characterizing the de- 
T... , t , ,v j . . , fj-i' formation of a continuous medium. 

condition that the determinant of this 

homogeneous system shall vanish and, as in the analogous case of the 
stress tensor discussed in the preceding paragraph, it is easy to show that 
the three roots fix three principal axes which in general are mutually 
orthogonal. Along these principal axes of strain, and along them only, 
the deformation consists of a pure stretching. The nature of the deforma- 
tion along any other axes can be visualized with the aid of a strain quadric 
such as (23). Let the point Pi, Fig. 14, have the coordinates 1, 2 , 8 
with respect to P and construct the surface 




(36) ^ = 5r 5s' = V Y a#fcfr = constant. 

/-I ffi 

If all lines issuing from P are extended, or if all are contracted, the quadric 
is an ellipsoid; if some lines are extended and others contracted, the sur- 
face is an hyperboloid. The relative displacement of PI with respect to 
P is given by 

(37) 5s' = iV*. 

The radius vector from P to PI on the surface S = 1 is 5r. In Fig. 15 
it has been assumed that this surface is an ellipsoid. The direction of 
the relative displacement 5s' due to the deformation is that of the normal 
erected at PI and its magnitude is equal to the reciprocal of the projection 



90 



STRESS AND ENERGY 



[CHAP. II 



NPi of the radius vector 5r on the normal. Obviously the vectors 5s' 
and 5r are parallel only along the principal axes of the ellipsoid. These 
axes represent also the directions of maximum and minimum contraction 
or extension. 

There is an interesting physical interpretation to be given to the 
tensor components a#. If before a deformation the relative position 
of two points P and Pi, Fig. 14, is fixed by 5r, their relative position 
after the deformation is 5r' = 5r + 5s', the prime over 5s' indicating 
that local rotations are excluded. Suppose now that 5r is directed along 

b 




FIG. 15. Graphical determination of strain at the point P from the strain quadric. 



the Zi-axis, so that 5x 2 = 5x 3 = 0. Then 5r' is a vector whose com- 
ponents are 

(38) 5xJ = (1 -f- aii) $Xiy 5x = ct 2 i 5xi, 5x 3 = a 3 i 5xi. 

The absolute value of the distance between P and PI after deformation is 

(39) |5r 7 | = V(l + an) 2 + ali + <*!i fai, 
and the relative change in length is 



(40) 



M - 



obtained by expanding (39) and discarding all terms of higher order 
than the first. Similar expressions can be found for deformations of line 
elements lying initially along the x 2 - and x s -axes, whence we conclude that 
the diagonal components a// = dsj/dx,- represent extensions of linear ele- 
ments which in the unstrained state are parallel to the coordinate axes. 

Again let 5a be a linear element which in the unstrained state is 
parallel to the i-axis and 5b be a linear element initially parallel to the 
a; 2 -axis. A local deformation transforms the vector 5a = ii 5zi into an 
element 



Sac. 2.2] ANALYSIS OF STRAIN 91 

while at the same time 6b = i 2 8x 2 is transformed into 
(42) 6b' = iiai 2 8x z + i 2 (l + a 22 ) 8x 2 + i 3 a 32 tei. 



In the initial state 6a and Sb are mutually orthogonal, but after the 
deformation, they form an angle which differs but little from ir/2 and 

which we shall denote by ^ 0i 2 . 

& 

(43) 3a' 8b' = |5a'| |5b'| cos fa - 0A 
But 

(44) cos f 1 - 0i 2 J = sin 12 ~ 6 l2j 

and, hence, 

(45) 0i 2 = ai 2 (l + an) + a 2 i(l + a 22 ) + a 3 ia 82 ; 
or, on neglecting terms of higher order than the first, 

(46) 12 = 0l2 + a21 = |!l + |??. 

0X2 oXi 

Thus, in general, the coefficient a#, with j ^ fc, measures one-half the 
cosine of the angle made by two linear dements after <i deformation, which 
in the unstrained state werz directed along a pair of orthogonal coordinate 
axes. 

Conforming to this interpretation, it is customary to write the com- 
ponents of the deformation tensor in a slightly altered manner. The 
coefficients defined by 

dsi dsi , ds 2 

11 = 3 #12 = -5 -r v~ = e 2 i 

dxi axz oxi 

/A7\ o $ s * * ds * i 5s s 

(47) e 22 = _ e = ~ + = e* z 



are called the components of strain, and the components of the relative 
displacement of any two points P and Pi due to a local deformation 
are therefore 

5si = en 8xi + |ei 2 8x 2 
(48) 5s 2 = ^e 2 i 8xi + 6 22 8x2 

xi + ^32 8x2 



If the origin of the arbitrary coordinate system Xi, rc 2 , x 3 is located at the 
point P and the axes then rotated into coincidence with the principal 
axes of the strain ellipsoid associated with that point, all components of 



92 



STRESS AND ENERGY 



[CHAP. II 



strain are reduced to zero with the exception of those lying on the main 
diagonal. We shall measure distances along the principal axes of strain 
by the coordinates u\, w 2 , u*. Then the components of any local deforma- 
tion with respect to P are 



(49) 

where 

(50) 



6s' 



= e 



e 2 = 



and where 8ui, 6u 2 , 6w 3 are the components of 5r with respect to the 
principal axes. The coefficients e\, e 2 , e 3 are called the principal strains? 
or principal extensions. 

Associated with the deformation of an infinitesimal region there is a 
change in the element of volume. If 6a, 5b, 8c are three vectors defining 
a parallelepiped, its volume is given by 

8ai 5a 2 

(51) 8V = (5a X 5b) 8c = 

Without loss of generality we may choose for this initial element a rec- 
tangular block whose sides are parallel to the coordinate axes. 

(52) 5a = ii 5zi, 5b = i 2 5x 2 , dc = i 3 8x s . 

The deformation transforms these vectors as in (41) and (42), whence 
for the volume after strain we find 

1 + 011 021 

012 1 + 022 



(53) 8V = (&*' X 5b r ) . 8c' 



8Xl 8X3 8X3. 



01S 2 3 1 + 033 

Expansion of this determinant and discard of all terms of higher order 
than the first lead to 

57' - 57 



(54) 



8V 



= an + 022 + 033 = V S 



for the change of volume per unit volume a quantity called the cubical 
dilatation. 

In summary, the analysis has shown that the most general displace- 
ment of particles in the neighborhood of a point may be resolved into a 
translation and a local rotation, upon which there is superposed a deforma- 
tion characterized by the strain components e^ and accompanied by a 
change in volume. The rotational component of the total strain can 
be induced only by rotational stresses force functions, that is to say, 
whose curl is not everywhere zero. 



Sac. 2.3] RELATIONS OF STRESS TO STRAIN 93 

2.3. Elastic Energy and the Relations of Stress to Strain. Under the 
Influence of external forces the particles composing an elastic fluid or 
solid suffer relative displacements. These strains, in turn, evoke internal 
forces which, in the case of static equilibrium, eventually compensate 
the applied stresses. The work done by the applied volume and surface 
forces in displacing every point of the medium an amount 6s is 

(55) W = J* f 5s dv + f t 5s da. 

According to Eq. (3) 

(56) t 5s = 6s x X n + 5s y Y n + ds t Z n, 
and, hence, 

(57) J* t 6s da = J V (X ds x + Y ds v + Z ds,) dv. 

Now 

(58) V (X ds x ) = texV X + X V ds x , 
and by (6) / = - V X. Thus (55) reduces to 

(59) dW = f (X V 8s x + Y V ds y + Z V 5s,) dv. 
The components of the integrand may, however, be written 

(60) X V *> - T () + T S (ft) + T 13 5 (), etc., 
which in the notation of Eqs. (47) leads to 

(61) tW = f (T n Sen + T 22 8e Z2 + T Z z 5e 33 + r 6e 12 + T 23 fe ss 

+ r 8 i tesi) dv 

for the work done by the applied stresses against the elastic restoring 
forces in the course of an infinitesimal change in the state of strain. If it 
be assumed that this change takes place so slowly that the variation in 
kinetic energy may be neglected, and that no heat is added or lost in the 
process, then SW must be equal to the increase in the potential energy U 
stored up in the elastically deformed medium. The elastic energy 
stored in unit volume will be denoted by w, whence 

(62) 6W = tU = Budv. 



The energy density u at any point must depend on the local state of 
strain and it may, therefore, be assumed that 

(63) u = u(e U) 612, e 83 ), 



94 SThESS AND ENERGY [CHAP. II 

or 



fdA\ XT7 C i dU * I dU H I dU 

(64) SU = J Us 5eu + ^ 5e * 2 + ^ 



*T 
063 

The strain components 6,-& are arbitrary; hence it follows that the com- 
ponents of applied stress (which in static equilibrium are equal and 
opposite to the induced elastic stresses) can be derived from a scalar 
potential function, 

(65) r . 

The existence of an elastic potential, which has been demonstrated 
here for the case of quasi-stationary, adiabatic deformations, may be 
shown for isothermal changes as well. 

When the deformations are not excessive, the components of elastic 
stress may be expressed as linear functions of the components of strain. 

TH = CiiBn + 612622 + 613633 + 614612 + 015623 + 615631, 

45623 



The coefficients c mn are called the elastic constants of the medium. The 
elastic potential u is in this case a homogeneous quadratic function of the 
strain components. The existence of a scalar function u(ej k ) satisfying 
(65) and (66) imposes on the elastic constants the conditions 

(67) C mn = Cnm (fl, W = 1, 2, , 6), 

whereby the number of parameters necessary to specify the relation of 
stress to strain in an anisotropic medium reduces from 36 to 21. Any 
further reduction is accomplished by taking advantage of possible sym- 
metries in the structure of the medium. If, in particular, the substance 
Is elastically isotropic, the quadratic form u must be invariant to orthog- 
onal transformations of the coordinate system, and the number of inde- 
pendent constants then reduces to two. The elastic potential assumes 
the form 

(68) u = i\i(cii + 622 
whence, by (65) 



(69) r 22 = Xi(6n + 6 22 + 6 33 ) + 2X 2 622, T 23 = X 2 e 23 , 

r 33 = Xifcu + 622 + 6 33 ) + 2x2633, TV = x 2 6 3 i. 



SBC. 2.3] RELATIONS OF STRESS TO STRAIN 95 

Now the conditions of static equilibrium are expressed by 
(16) f + V - 2 T = 0. 

Upon introducing the components (69) into the expanded divergence 
and expressing the strain components ,* in terms of the deformation s, 
we find 

(70) f + (Xi + X 2 )VV - s + X 2 V 2 s = 

as the equation determining the deformation of an isotropic body sub- 
jected to a volume force f. The constants of integration are evaluated 
in terms of data specifying either the displacement of points on the 
bounding surface, or the distribution of stress over that surface. 

The parameters Xi and X 2 are positive quantities. X 2 is called the 
rigidity, or shear modulus, for it measures the strains induced by tan- 
gential, or shearing, stresses. No simple physical meaning can be 
attached to Xij however, it may be defined in terms of the better known 
constants E and a, where Young 1 s modulus E is the ratio of a simple 
longitudinal tension to the elongation per unit length which it produces, 
and where the Poisson ratio <r is the ratio of lateral contraction to longi- 
tudinal extension of a bar under tension. Then 

E<r E 



01 (1+cOtt -2cr)' /v * 2(1+ a) 

An ideal fluid supports no shearing stress, and hence X 2 = 0. The 
stress acting on any element of a closed surface within the fluid is, there- 
fore, a normal pressure, 

(72) Tn = T 22 = T 33 = -p, T w = T 23 = T 31 = 0, 

the negative sign indicating that the stress is directed inward. The 
components of strain en, 622, 33 are all equal and 

(73) p = Xi(cn + <? 22 + e 33 ) = XiV s, 
while the equation of equilibrium reduces to 

(74) f - Vp = 0. 

Let Fo be the initial volume of an element of fluid, V\ its volume at a 
pressure pi, and F 2 its volume at a pressure p 2 . From the definition 
(54) of the cubical dilatation V s we have 

T7 -\T T7 T7 

(75) p, = -Xi- 



and for sufficiently small changes, 
(76) p,-pi= 



96 STRESS AND ENERGY [CHAP. II 

For an infinitesimal change in pressure 

/i-,\ j x dV x dr 

(77) dp = -Xi-y = Xi , 

where T is the density of the fluid. The reciprocal of Xi is in this case 
called the compressibility. 

ELECTROMAGNETIC FORCES ON CHARGES AND CURRENTS 

2.4. Definition of the Vectors E and B. An electromagnetic field is 
defined according to our initial hypotheses by four vectors E, B, D and H 
satisfying Maxwell's equations. The physical nature of these vectors 
will be expressed in terms of experiments by means of which they may be 
measured. Now it is easy to show that pE and J X B are quantities 
whose dimensions are those of force per unit volume. For by Sec. 1.8 

(1\ f pi c ul m bs v volts __ kilograms 

meter 3 meter second 2 meter 2 

/ox TT ^ -01 amperes . . webers kilograms 

(2) U X BJ = X - 



meter 2 meter 2 second 2 meter 2 

We are, therefore, free to define the vectors E and B as forces exerted 
by the field on unit elements respectively of charge and current. More 
precisely, we shall suppose that charge is distributed throughout a volume 
V with a macroscopically continuous density p. Then E is defined such 
that the net mechanical force acting on the charge is 

(3) F e = f yP Edv, 

distributed with a volume density 

(4) f. = P E. 

If V is sufficiently small, the field within this region at any instant may 
be assumed homogeneous, so that in the limit 

(5) F e = 

In like manner the association of the magnetic vector B with the 
force exerted on a unit volume element of current through the equation 

(6) f m = J X B, 

leads to 

(7) 

as the net force exerted on a volume distribution of current. If the 
current is confined to a linear conductor of sufficiently small cross section, 



Sue. 2.5] 



THE ELECTROMAGNETIC STRESS TENSOR 



97 



B may be assumed homogeneous and the current lines parallel to an 
element of length ds. The force acting on a linear current element is then 



(8) 



d m = I cfe X B. 



dF m 



Whereas the force exerted by an electric field on an element of charge is 
directed along the vector E, the force exerted on an element of current 
in a magnetic field is normal to the plane defined by the element ds 
and the vector B. 

This arbitrariness in our definition of E and B is inevitable. The 
mutual forces exerted by charges upon charges or currents upon currents 
can be measured, but the field vectors themselves are not independent 
entities accessible to direct observation. 
The definitions of E and B based on Eqs. 
(3) and (7) have been shown by purely 
dimensional considerations to be compat- 
ible with Maxwell's equations. In the 
following we shall have to show that the 
properties of a field of the vectors E and B 
defined in this manner and satisfying 
these equations are in complete accord 
with experiment. From the forces ex- 
erted by charges and currents may be 
determined the work necessary to estab- 
lish a field; from energy relations in turn it will be possible to deduce the 
forces exerted on ponderable elements of neutral matter. 

2.5. The Electromagnetic Stress Tensor in Free Space. Let us sup- 
pose that a certain bounded region of space contains charge and current 
distributions but is free of all neutral dielectric or magnetic materials. 
The field is produced in part by the charges and currents within the 
region, in part by sources which are exterior to it. At every interior point 




FIG. 16. Direction of force exerted 
on a current element J ds in a mag- 
netic field B. 



(I)VXE + = 0, 



(III) V B = 0, 



r 

(II) V X B - ^o = MoJ, (IV) V . E = -p. 

Let (I) be multiplied vectorially by oE, (II) by the vector B. Upon 
adding and transposing terms we find 



(9) e(V X E) X E + (V X B) X B = J X B + eo (E X B). 

Mo <n 

In a rectangular system of coordinates the first term of (9) may ba 
represented by the determinant 



98 



STRESS AND ENERGY 



[CHAP. II 



(10) (V X E) X E = 



j k 

x dE f dE v 



~dy 



Ey 



Eg 



The ^-component of this vector is 



(11) 



dx 



dy 



s '~dF 



V 
Jli v 

v 



""to" 

\ TTT a T}1 

O-C/y J-, CfjUz 

dx * dx 



Now the quantities 
(12) Sft 



transform like the components of a tensor [Eq. (43), page 66] and the 
first three terms on the right of (11) constitute therefore the x-component 
of the divergence of a tensor 2 S (a) . The remaining components are 
calculated from the T/- and z-components of c (V X E) X E, such that 
we are led to the identity 

(13) (V X E) X E = div 2 S<*> - c E V E, 

the components of 2 S (6) being tabulated below. 

TABLE I. COMPONENTS S ( $ OF THE TENSOR 2 S (fl) IN FREE SPACE 



The transformation of (V X B) X B is effected similarly, giving 



(14) 



~ (V X B) X B = div 2 S<*> - - B V 
Mo Mo 



where the components of 2 S (ns) are as represented in Table IT. 



SBC. 2.5] THE ELECTROMAGNETIC STRESS TENSOR <J9 

TABLE II, COMPONENTS S ( /f OF THE TENSOR 2 S (m) IN FREE SPACE 



^v^ 


1 


2 


3 


i 


Mo * 2Mo 


B X B V 
Mo 


1 




1 


1 1 


1 


2 


ByBx 


J$* j^2 






Mo 


Mo v 2Mo 


Mo 




1 


1 


1 i 


3 


MO * * 


B Z B V 

Mo v 


Mo * 2/xo 



Upon replacing the first two terms of (9) by (13) and (14) and taking 
account of (III) and (IV), we obtain an identity of the form 



(15) 



div 2 S = Ep + J X B + e (E X B), 



wherein the components of the tensor 2 S are 

(16) s* = stf + sj?>, 

and where 



, 3 3 
div 2 S = ' >, >, i/ 



(17) 



Equation (15) is a relation through which the forces exerted on elements 
of charge and current at any point in otherwise empty space may be 
expressed in terms of the vectors E and B alone. 

Let us integrate this identity over a volume V. Now the integral 
of the divergence of a tensor throughout V is equal to the integral of a 
vector over the surface bounding V. To demonstrate this tensor ana- 
logue of the vector divergence theorem, let n be the outward unit normal 
at a point on the bounding surface and consider Su to be the x-component 
of a vector whose y- and ^-components are zero. The component of this 
vector in the direction of the normal n is Snn X) whence by the divergence 
theorem 



(18) 



Jr ^>s 
Sun, da = j -^ dv. 



Likewise it is apparent that 

J 
/ 



(19) 



- f f^> 

J d/ ' 

-/ 



r 



100 STRESS AND ENERGY .,HAP. II 

Upon adding these three identities, we obtain 

120) J (flun. + &*. + SnJ da = J (*.' + jbs + ***) dv . 

The integrand on the right is the ^-component of div 2 S, and consequently 
(21) *, 



is the x-component of a vector t which according to (20) is to be integrated 
over the surface bounding V. Proceeding similarly for the other com- 
ponents, we have the general theorem: 



(22) t da = div 2 S dv, 
where the components of t are 

(23) t, = 2 S*n, (j = 1, 2, 3), 



or, in abbreviated notation, 

(24) t = 2 S n. 

Applied to Eq. (15), the divergence theorem (22) leads to 

(25) J 2 S - n da F, + F m + o ~ J E X B dv 

with F and F m representing, as in Eqs. (3) and (7), the resultant forces 
acting respectively on the charge and the current contained within V. 

Consider first stationary distributions of charge and current. The 
fields are then independent of time and the third term on the right of (25) 
is zero. Equation (25) now states that the force exerted on stationary 
charges or currents can be expressed as the integral of a vector over any 
regular surface enclosing these charges and currents. It does not state 
that the volume forces F and F m are maintained in equilibrium by the 
force 2 S n distributed over the surface. The equilibrium must be 
established with mechanical forces of some other type, and in fact it will 
be shown shortly that a charge distribution cannot possibly be maintained 
in static equilibrium under the action of electrical forces alone. To be 
more specific, let us imagine a stationary charge distribution to be divided 
into two parts by an arbitrary closed surface S. The force exerted by 
the external charges on those within S must in some manner be trans- 
mitted across this surface. The net force on the interior charges may, 
according to (25), be correctly calculated on the assumption that the 
force transmitted across an element of area da is 2 S (e> n da, and the 
components SJp are hence effective stresses in the electrostatic field. 



SEC. 2.5] THE ELECTROMAGNETIC STRESS TENSOR 101 

Associated with every point in the field there is a stress quadric from 
which may be determined the normal and tangential components of 
transmitted stress on an element whose normal is n. To find the prin- 
cipal axes of this quadric with respect to the direction of the field we shall 
adopt the procedure described in Sec. 2.1. The secular determinant (19) 
of that section assumes in the present instance the form 



(26) 



= 0. 



- X 

- X 

o(#* - %E 2 ) - X 

When expanded and reduced by taking account of the relation 

E* + El + El = # 2 , 
Eq. (26) proves equivalent to 

(27) - 8X 3 + 4JS7 2 X 2 - 2S 4 X - JS 6 = 0. 
The roots of (26) are, therefore, 

(28) X a = |tf>, X 5 = X = -|JP, 

from which it is apparent that the stress quadric has an axis of symmetry. 
Let n (o) be a unit vector fixing the direction of the principal axis 
associated with X . According to (18) page 86, the components of 
n (o) with respect to an arbitrary reference system must satisfy 

(El - E*)n? + EJB v ny> + E*E z n = 0, 

(29) EJEjtf + (El - E*}nf + E v E z n<?> = 0, 

- E*)n = 0. 



From the theory of homogeneous equations it is known that the ratios of 
the unknowns n ( x a \ n ( \ n 0) are as the ratios of the minors of the deter- 
minant of the system, whence one may easily verify from (29) that 

(30) nL 0) : <> : n<* = E x : E v : E t . 

The major axis of the electric stress quadric at any point in the field is directed 
along the vector E at that point. The stress transmitted across an element 
of surface whose normal is oriented in this direction is a simple tension, 

(31) t<> -|j57 2 n^). 

The stress across any element of surface containing the vector E t.e v , 
an element whose normal is at right angles to the lines of force is also 
normal but negative, and corresponds therefore to a compression 

(32) t< 6 > =- - 



102 



STRESS AND ENERGY 



[CHAP. II 



Suppose, finally, that the normal of a surface element in the field ig 
oriented in an arbitrary direction specified by n. Let the z-axis of a 
coordinate system located at the point in question be drawn parallel to E 
and choose the z-axis perpendicular to the plane through E and n. The 
angle made by n with the direction of E will be called 0. Then 

E x = E y = 0, IEI = E g , 
n x = 0, n y == sin 0, n z = cos 0, 

whence according to Eq. (23) the stress components are 



(33) 



0, 



sin 0, t z = 2 cos Q. 



The absolute value of the stress transmitted across any surface element, 
whatever its orientation, is therefore 

(34) |t|=!# 2 . 

Furthermore t lies in the plane of E and n in a direction such that E 

bisects the angle between n and t as illu- 
strated in Fig. 17. 

In the light of this representation it is 
easy to comprehend the efforts of Faraday 
and Maxwell to reduce the problem of elec- 
tric and magnetic fields of force to that of 
an elastic continuum. To both, the con- 
cept of a force propagated from one point in 
space to another without the intervention 
of a supporting medium appeared wholly 
untenable, and in the absence of anything 
more tangible an all-pervading " ether" 
was eventually postulated to fill that role. 
There was then attributed to the stress 
components of the field, even in space free of matter, a physical 
reality, and a valiant attempt was made to associate with the ether 
properties analogous to the strains of elastic media. These efforts did 
not bear fruit. Subsequent research has shown that electromagnetic 
phenomena may be formulated without employment of any fixed refer- 
ence system, and that there are no grounds for the assumption that force 
can be propagated only by actual contact of contiguous elements of mat- 
ter or ether. On the modern view, the representation of an electrostatic 
field in terms of the stress components S$ has no essential physical 
reality. All that can be said and all that it is necessary to say is 
that the mutual forces between elements of charge can be correctly 




FIG. 17. Relation of tension t 
transmitted across an element of 
surface in an electrostatic field to 
the field intensity E. 



SBC. 2.6] ELECTROMAGNETIC MOMENTUM 103 

calculated on the assumption that there exists throughout the field a 
fictitious state of stress as described. 

Since the tensors 2 S (m) and 2 S (e) are of identical form, the discussion of 
the foregoing paragraphs applies in its entirety to the field of a stationary 
distribution of current. The force exerted by external sources on the 
current within an arbitrary closed surface is obtained by integrating over 
this surface a stress whose absolute value is 

(35) |t| = B*, 

*Mo 

and whose direction with respect to the orientation of a surface element 
and the direction of the field B is as in Fig. 17. 

If in Eq. (23) the components Sjk are introduced from Tables I and II, 
it is apparent that the stress transmitted across an element of area whose 
positive normal is n may be written vectorially as 

(36) t<<> = 2 S> n = c (E n)E - ^ 

& 

(37) t< m) = 2 S<" n = - (B n)B - ^- 

Mo ^Mo 

Over any volume bounded by a regular, closed surface S we have there- 
fore for stationary distributions of charge and current: 



(38) 
(39) 



I e (E n)E - | E*n 1 da = f P E dv, 

- (B n)B - ^ J3 2 n 1 da = f J k B 
MO 2/10 J Jv 



dv. 



2.6. Electromagnetic Momentum. In a stationary field the net force 
transmitted across a closed surface S bounding a region containing neither 
charge nor current is zero. If, however, the field is variable, it is clear 
from Eq. (25) that this is not the case. How, then, are we to interpret 
the apparent action of a force on volume elements of empty space? 
Since the dimensions of e E = D are QL~ 2 , and those of B = /* H are 
7 "" 1 , it is evident that the quantity 



(40) g = 6 ExB = ~ExH 

c 

is dimensionally a momentum per unit volume. The dimension Q drops 
out of the product D X B and this conclusion, therefore, is not simply a 
consequence of the particular system of units employed. The identity 

(41) 



104 STRESS AND ENERGY [CHAP. II 

can be interpreted on the hypothesis that there is associated with an 
electromagnetic field a momentum distributed with a density g. The 
total momentum of the field contained within V is 



(42) 



= I g dp kg.-meters/sec., 

\) * 



and (41) now states that the force transmitted across 2 is accounted for 
by an increase in the momentum of the field within 2. The vector 
2 S n measures the inward flow of momentum per unit area through 2, 
while the quantity Sjk may be interpreted as the momentum which in 
unit time crosses in the j-direction a unit element of surface whose normal is 
oriented along the k-axis. 

A direct consequence of this hypothesis is the conclusion that New- 
ton's third law and the principle of conservation of momentum are 
strictly valid only when the momentum of an electromagnetic field is 
taken into account along with that of the matter which produces it. Let 
us suppose that within the closed surface 2 there are charges distributed 
with a density p, and that the motion of these charges may be specified 
by a current density J. The force exerted on the charged matter within 
2 is then 

(43) F. + F m = f ( P E + J X B) dv - ^ < 

J at 

where G meo h is the total linear momentum of the moving, ponderable 
charges. The conservation of momentum theorem for a system com- 
posed of charges and field within a bounded region is, therefore, expressed 
according to Eq. (25) by 

(44) ~ (G me0 h + G olectroma .) = f 23 . n da. 
at Js 

If the surface 2 is extended to enclose the entire field, the right-hand 
side of (44) must vanish, and in this case 

(A K\ /"^ I f\ 

\*^ ) ^^rocoh I vj<ii Kt.rr>m ^ ~~~ constant. 

There appears to be associated with an electromagnetic field an inertia, 
property similar to that of ponderable matter. 

ELECTROSTATIC ENERGY 

2.7. Electrostatic Energy as a Function of Charge Density. A finite 
charge concentrated in a region so small as to be of negligible extent 
relative to other macroscopic dimensions will be referred to as a point 
charge. Now the force exerted on such a point charge q in the field of a 
stationary charge distribution is gE, and the work done in a displacement 



SBC. 2.7] CHARGE DENSITY 105 

of q from a point r = TI to a second point r = 12 is 

(1) W = g f r ' E ds. 

/ri 

Since the curl of E vanishes at every point in an electrostatic field, the 
vector E is equal to the negative gradient of a scalar potential <, and we 
have 

(2) E rfs = - V0 - ds = -d0, 

where d<t> is the change in potential along an element ds of the path of 
integration. From this it follows clearly that the work done in a dis- 
placement of q from ri to r 2 is independent of the choice of path and is a 
function solely of the initial and terminal values of the potential. 

(3) W = - 

* 

In particular, the work done in the course of a displacement about a 
closed path is zero. A field of force is said to be conservative if the work 
done in a displacement of a system of particles from one configuration 
to another depends only on the initial and final configurations and is 
independent of the sequence of infinitesimal changes by which the finite 
displacement is effected. The conservative nature of the electrostatic 
field is established by (3), or more precisely by the condition V X E = 0. 
Displacements and variations in a static field must be understood to occur 
so slowly as to be equivalent to a sequence of stationary states. Such 
changes are said in thermodynamics to be reversible. 

In Chap. Ill it will be shown that if all the sources of an electrostatic 
field are located at finite distances from some arbitrary origin the poten- 
tial and field intensities become vanishingly small at points which are 
sufficiently remote. The work done by a charge q as it recedes from an 
initial point r = TI to r 8 = oo is, therefore, 



(4) W = 

Obviously the scalar potential itself may be interpreted as the work done 
against the forces of the field in bringing a unit charge from infinity to 
the point r, or as the work returned by the system as a unit element of 
charge recedes to infinity. 

(5) <t>(x, y, z) = -f^ E . ds. 

We shall use the term energy of an electrostatic system somewhat 
loosely to mean the work done on the system in carrying its elements of 
charge from infinity to the specified distribution by a sequence of reversi- 



106 STRESS AND ENERGY [CHAP. H 

ble steps. It will be assumed that the temperature of all dielectric or 
magnetic matter in the field is held constant. 1 

The energy of a point charge 2 in the field of a single point source q\ is 

(6) U = 02021, 

where 2 i is the potential at 2 due to 0i. Now the work done in bringing 
02 from infinity to a terminal point in the field of q\ would be returned 
were q\ allowed next to recede to infinity, and therefore 

(7) U = 0i0i2. 

The mutual energy of the two charges may consequently be expressed by 
the symmetrical relation 

(8) U = 4(01012 + 02021). 

If first 02 and then 3 be introduced into the field of 0i, the energy is 

(9) U = 02*21 + 03(031 + 032), 

which in virtue of the reciprocal relations between pairs is equivalent to 

(10) U = 4(012 + 013)01 + 4(021 + 023)02 + 4(031 + 03 2 )03. 

By induction it follows that the energy of a closed system of n point 
charges is 

n n n 

(n) u = 1 2 2 *** = 1 2 **' 

t=l ;1 =! 

tyy 

where is the potential at 0t due to the remaining n 1 charges of the 
system. 

Note that (11) is valid only if the system is complete, or closed. If 
on the contrary the n charges are situated in an external field of potential 
0o, a term appears which does not involve the factor 4- In this case 



Let us consider now a region of space V within which there are to be 
found fixed conductors and dielectric matter. Within the dielectrics 
charge is distributed with a volume density p, while over the surfaces of 
these dielectrics and on the conductors there may be thin layers of charge 
of surface density o>. At a point (x, y, z) within V the potential of the 
distribution plus that of possible sources situated outside V is 0. The 
work required to increase the charge at (x, y, z) by an infinitesimal 

1 The energy in question is in fact the free energy hi a thermodynamic sense. 



SBC. 2.8] FIELD INTENSITY 107 

amount 5q is < Sg, and the increase in energy resulting from an increase 
in the volume density of charge by an amount Sp, or the surface density 
by an amount fao, at every point in V is 

(13) dUi = J" <f> dp dv + C <t> 5co da. 

The second integral is to be extended over all surfaces within V bearing 
charge. On the other hand, the addition of an element of charge 6q at 
(x, y, z) increases the potential at all points both inside and outside V 
by an amount 6<, with a resultant increase of the energy of the charges 
already existing within V of 

(14) SC/2 - J* p &<f> dv + f o> S<t> da. 

The work 5[/i done on the system in building up the charge density is 
equal to the potential energy dU 2 stored in the field provided the system 
is closed; provided, .that is, that the region V of integration is extended 
to include all charges contributing to the field. In that case 

SU = SUi = 6C7 2 , 

(15) BU - \ J (* 8p + p 50) dv + ~ J (* du + co 40) da, 
which upon integration leads to 

(16) V = \ 

as the electrostatic energy of a charge system referred to the zero state 
p = w = o. 1 

2.8. Electrostatic Energy as a Function of Field Intensity. Let us 
imagine for a moment that conductors have been eliminated from the fie!d 
and that all surface discontinuities at the boundaries of dielectrics are 
replaced by thin but continuous transition layers. Charge is distributed 
throughout the dielectric with a density p, but we shall assume that this 
distribution is confined to a region of finite extent: the potential and 
intensities of the field vanish at infinity. Regions free of matter are, of 
course, to be considered as dielectrics of unit inductive capacity. The 
work required to increase the charge density at every point in the field 
by an amount dp is 

(17) SU = f <t> Bp dv, 

where </> is the potential due to the initial distribution p. Now the 
increment in charge density is related to a variation of the vector D 

1 The convergence of these integrals will be demonstrated in Chap. III. 



108 



STRESS AND ENEMY 



[CHAP. II 



D) = V (SD). 



by the equation 

(18) Sp = 
Furthermore, 

(19) <f>V - (SD) = V - (0 SD) - SD V< = V (0 SD) + E 
and hence (17) is equivalent to 



(2t; 6C7 = E - 5D dv + V - (<f> SD) dv. 

or, upon application of the divergence theorem, to 
(21) dU = E SD dv + <t> 5D n da, 



where S is a closed surface bounding a volume V. This region V need 
not include all the charges that contribute to the field. If, however, we 

allow the surface S to expand into 
a sphere of infinite radius about 
some arbitrary origin, the contri- 
bution of the surface integral 
vanishes ; for <t> will be shown later 
to diminish as 1/r at sufficiently 
large distances from the origin, 
and D as 1/r 2 . The surface S in- 
creases with increasing radius as 
r 2 , and the surface integral there- 
fore vanishes as 1/r. The incre- 
ment of energy stored in the 
electrostatic field can be calculated 
from the integral 

FIG. 18. Conductors bounded by the (22) dU = f E SD dv 

surfaces Si, St, . . . , embedded in a ^ ' J 

dielectric medium. , i j n 

extended over all space. 

In practice the charge is rarely distributed throughout the volume of a 
dielectric but is spread in a thin layer of density w over the surfaces of 
conductors. These conductors may be considered as electrodes of 
condensers, and the increase in the energy of the field is the result, for 
example, of work done by the electromotive forces of batteries in the 
process of building up the charge on the electrodes. Let us suppose that 
there are n conductors, whose surfaces we shall denote by /Si, embedded 
in a dielectric of infinite extent. One of these surfaces, let us say S n , will 
be assumed to enclose all the others as illustrated in Fig. 18. To ensure 
the continuity of the potential and field necessary for the application of 
the divergence theorem, surfaces of abrupt change in the properties of the 
dielectric may again be replaced by thin transition layers without in 




SEC. 2.8] FIELD INTENSITY 109 

any way affecting the generality of the results. The work required to 
increase the charge density on the surfaces Si by an amount $u> is 

(23) SU = jj f s <t> 5co da,i. 

The sum of these n surface integrals can be represented as a single integral 
by the artifice of drawing finlike surfaces from the interior conductors to 
S n as indicated in Fig. 18. The integration starts at a point P on S n , 
extends on one side of the fin to Q, then over Si and then returns to P 
over the opposite face of the fin. The surface PQ carries no charge and 
contributes nothing to the integral. The interior of S n is thus reduced 



to a simply connected region 1 bounded by a single surface S = ^ Si. 

t i 

(24) dU = f <t>duda. 

js 

In Sec. 1.13 it was shown that at any surface of discontinuity the 
normal and tangential components of the vectors D and E satisfy the 
conditions 

(25) n (D 2 - 7>i) = w, n X (E 2 - EI) = 0. 

The positive normals to the surfaces Si are directed into the dielectric, 
and since n in Sec. 1.13 was drawn from medium (1) into medium (2), 
the index (1) now denotes the interior of the conductors whereas (2) 
refers to the dielectric. In Chap. Ill we shall have occasion to consider 
in some detail the electrostatic properties of conductors, but for the 
present we need only accept the elementary fact that the field at any 
interior point of a conductor is zero. Were it otherwise, a movement of 
free charge would occur, contrary to the assumption of a stationary state. 
The interior and surface of a conductor are, therefore, a region of constant 
potential. The charge density at interior points is zero [Eq. (21), page 
15] and whatever charge the conductor carries is distributed on the 
surface in such a way as to bring about the vanishing of the interior field. 
Since EI = DI = 0, 

(26) n-D 2 = w, n X E 2 = 0; 

at a point just outside the conductor the tangential component of E 
is zero and the normal component of D is equal to the surface density 
of charge. 

Upon introducing the first of these relations into (24) and dropping 
the index, we obtain 

(27) dU = f <MD-nda. 

js 

1 See Sec. 4.2, p. 227. 



HO STRESS AND ENERGY [CHAP. II 

To express this result as an integral extended throughout the volume of 
dielectric bounded by the surfaces Si we again apply the divergence 
theorem, but must note that n in (27) is directed into the dielectric; the 
sign must therefore be reversed. 

(28) SU - f V fa 3D) dv = J*E 3D dv - J <V (3D) <fe. 

If the density of charge throughout the dielectric is zero or constant, 
V (3D) = 0, and we find as in (22) that the work expended by the 
batteries in building up the electrode charges by an amount 5co is 

(29) 317 = J E 3D dv, 

an integral extended over the entire space occupied by the field, and that 
this work is stored in the field as electrostatic energy. 

Since the energy 5C7 appears to be stored in the field, as the potential 
energy of an extended spring is in some manner stored within it, it is 
not unreasonable to suppose that the electrostatic energy is distributed 
throughout the field with a density 

(30) du = E 3D. joules/meter 3 

It is difficult either to justify or disprove such a hypothesis. The 
transformation from a surface integral to a volume integral is obviously 
not unique, for there might be added to < 3D in (27) any vector whose 
normal component integrated over S is zero. Furthermore, it may be 
questioned whether the term "energy density" has any physical sig- 
nificance. Energy is a function of the configuration of a system as a 
whole. The objection has been stated in a rather quaint way by Mason 
and Weaver, 1 who suggest that it is no more sensible to inquire about the 
location of energy than to declare that the beauty of a painting is dis- 
tributed over the canvas in a specified manner. However ingenious, 
such an analogy seems not entirely well founded. The energy of an 
inhomogeneously stressed elastic medium is certainly concentrated 
principally in regions of greatest strain and in this case the elastic energy 
per unit volume has a very definite physical sense. Granting that the 
analogy of the electrostatic to the elastic field is not a close one, and that 
we can be certain only of the correctness of the expression (29) integrated 
over the entire field, it is nevertheless plausible to assume that the 
energy is localized in the more intense regions of the field in the manner 
prescribed by (30). 

To find the total energy stored in a field, the increment dU must be 
integrated from the initial state D = to the final value D. 

1 MASON and WEAVEB, "The Electromagnetic Field," p. 266. University < I 
Chicago Press, 1929. 



SBC. 2.9] A THEOREM ON VECTOR FIELDS 111 

(31) U 



In case the medium is isotropic and linear, such that c in the relation 
D = eE is a function possibly of position but not of E, we have 

E 5D ~ SE 2 , 
j 

and hence 

(32) t/ = i f cE 2 dy. 

2.9. A Theorem on Vector Fields. Let P and Q be two vector func- 
tions of position which throughout all space satisfy the conditions 

(33) V X P = 0, v Q = 0, 

and which are continuous and have continuous derivatives everywhere 
except on a closed, regular surface Si. The transition of the tangential 
components of P and of the normal components of Q across the surf ace Si 
is assumed continuous, but arbitrary discontinuities are permissible 
in the normal components of P and the tangential components of Q. 
The prescribed conditions over Si are, therefore, 

(34) nx(P + -P_) = 0, n.(Q + -Q_) = 0. 

The unit normal n is drawn outward from Si and vectors in the immediate 
neighborhood of this outer face are denoted by the subscript +, whereas 
those located just inside the surface are denoted by the subscript . 
Finally, it is assumed that the sources of the fields P and Q are located 
at finite distances from an arbitrary origin and that P and Q vanish at 
infinity such that 

(35) lim rP = 0, lim rQ = 0. 

r > r 

Then k can be shown that the integral over all space of the scalar product 
of an irrotational vector P and a solenoidal vector Q is zero, provided P and 
Q and their derivatives are continuous everywhere except on a finite number 
of closed surfaces across which the discontinuities are as specified in (34). 
For since V X P = 0, we may write P = V<t> and, hence, 



(36) P Q = - V (4>Q) + 0V Q. 

We shall denote the volume enclosed by the surface Si by Vi, and that 
exterior to it by F 2 , so that the complete field of P and Q is Vi + F 2 . 
The last term of (36) vanishes and therefore 



(37) / Fi+rj P - Q dv = - f vi V - (Q) dv - J F| V (*Q) dv. 

To apply the divergence theorem to this expression we observe that 



112 STRESS AND ENERGY [CHAP. II 

is bounded by Si, and that V* is bounded on the interior by Si and on the 
exterior by a surface S which recedes to infinity. In view of the behavior 
of P and Q at infinity as specified by (35), the integral of <t>Q n over S is 
zero. The positive normal to a closed surface is conventionally directed 
outward from the enclosed volume; the two integrals of <Q n over S\ 
are therefore of opposite sign but of equal magnitude in virtue of (34). 
Equation (37) transforms to 

(38) h+v> P ' Q *> = -f Sl *Q+ ' n da + f Sl *Q- ' n da = > 

as stated. The extension of this theorem to a finite number of closed 
surfaces Si of discontinuity is elementary. 

2.10. Energy of a Dielectric Body in an Electrostatic Field. The 
useful theorem of the preceding section may be applied to the solution of 
the following problem. Let us suppose that an electrostatic field EI has 
been established in a dielectric medium. To simplify matters we shall 
assume that this medium is isotropic and linear } and hence characterized 
by an inductive capacity i which is either constant or at the most a 
scalar function of position. A nonconducting body whose inductive 
capacity is 2, is now introduced into the field EI, while the sources of EI 
are maintained strictly constant. We wish to know the energy of the 
foreign dielectric body due to its position in the field. 

The initial energy t/i, representing the total work done in establish- 
ing the initial field, is obtained by evaluating 



= \ J 



(39) Ui = Ei D! dv 

over all space. After the introduction of the body the modified field at 
any point is E, and the difference E 2 = E EI is thus the field resulting 
from the polarization of the body. The volume occupied by the foreign 
body we denote by V\ y that of the medium exterior to it by V*. The 
energy of the field in this new state is 

(40) V* = | f E D dv, 

* jFi-f 7i 

and the change 

(41) U = f/ 2 - Ui = J f (E D - Ex Di) dv 

* JVi+Vt 

must be the energy of the body in the external field EI, and consequently 
equal to the work done in introducing it. Equation (41) is equivalent to 



(42) 



U = i f E - (D - Di) dv + ~ f (E - EI) - D x dv. 

* JVi+Vt * JVi+Vt 



SBC. 2.10] ENERGY OF A DIELECTRIC BODY 113 

The curl of E is zero ev?ry where, and the divergence of D DI is zero 
since the initial source distribution is fixed. Across the surface which 
bounds the medium 2 from the medium i the conditions 

(43) n X (E+ - E_) =0, n - [(D - DO+ - (D - DO-] = 0, 

are satisfied provided the surface carries no charge. Then by the theorem 
of Sec. 2.9 the first integral of (42) is zero, leaving 

(44) V = 5 f (E - Ei) D! dv + 1 f (E - Ei) - D! dv. 

* JVi 4 JV* 

Since DI = iEi, the second integral of (44) is equivalent to 

(45) 5 f (B - Ei) Di cfo = 5 f (D - DO - EX dv. 

* JV* * JVi 

The conditions of Sec. 2.9 are satisfied by V X EI = 0, V (D DI) = 0, 
so that 

(46) f E! (D - DO dv ~ f Ei (D - DO dv 

JVi+Vi JVi 

+ I Ei (D - DO dv = 0, 

J Vt 

and hence 

(47) 5 f E! (D - DO dv = -5 f E! - (D - DO dv. 

^ JVt * JVi 

Upon introducing (47) into (44), we obtain an expression for the energy 
of a dielectric body embedded in a dielectric medium in terms of an 
integral extended, not over all space, but over its own volume alone. 



(48) C7 = ^J^(E 2 .D 1 ~E 1 -D 2 )&, 
or, since DI = cJSi, D = e 2 E within Vi, 

(49) U = i f (E D! - Ei D) dv = f ( l - e 2 )E . E l dv. 

* JVi * JVi 

In case the external medium is free space, the inductive capacity j 
reduces to c . Then, since the resultant field within the body is related 
to its polarization according to Sec. 1.6, by 

(50) D = E + P 2 E, P = ( 2 - o )E, 
U may be written 



(51) U 



= ~i f P Ei dv. 

* JVi 



114 STRESS AND ENERGY [CHAP. II 

The potential energy of a dielectric body in free space and in a fixed external 
field EI is equal to P EI per unit volume. 

It is the variation of this potential energy U expressed by (49) or (51) 
with respect to a small displacement that will lead us to the mechanical 
forces exerted on polarized dielectrics. It will be observed that the 
energy is negative if e 2 > ei, and decreases in this case with decreasing ei, 
increasing e 2 , or increasing field intensity EI. We may anticipate, there- 
fore, that if 2 > 61, the body will be impelled toward regions of intense 
field or diminishing inductive capacity 1. On the other hand if 1 > e 2 , as 
might be the case of a solid immersed in a liquid of high inductive capac- 
ity, the forces exerted on the body will tend to* expel it from the field. 

A direct consequence of (49) is the theorem that any increase in the 
inductive capacity of a dielectric results in a decrease in the total energy of 
the field. Let us suppose again that EI is the field of & fixed set of charges 
in a dielectric medium whose inductive capacity is e = t(x, y, 2). If a,t 
every point is increased by an infinitesimal amount 5e, the consequent 
variation in the electrostatic energy will according to (49) be equal to 



_ 1 f 
~~2j 



(52) dU = -^ I 5tE*dv, 

for the product deE EI will then differ from Be E 2 by an infinitesimal of 
second order. 

The energy of an electrostatic field is now completely determined by 
the distribution of charge p and w, and by the inductive capacity e(x, y, z). 
Equation (52) expresses the variation in energy resulting from a slight 
change in the properties of the dielectric, in the course of which the 
charges are held constant. 

(53) U = -H I I E *dedv, (constant charge). 
The variation 

(54) 8U = I <t> 8p dv + I Sco da = j E - SD dv 

expresses, on the other hand, the increment of energy resulting from a 
small change in the density of charge, in the course of which the proper- 
ties of the dielectric are held constant. 

2.11. Thomson's Theorem. Charges placed on a system of fixed con- 
ductors embedded in a dielectric mil distribute themselves on the surfaces of 
these conductors such that the energy of the resultant electrostatic field is a 
minimum. 

The proof of this and the following theorems will be confined to the 
case of a linear, isotropic dielectric. Let us suppose that there are n 



SEC. 2.11] THOMSON'S THEOREM 115 

conductors bounded by the surfaces S, (i = 1, 2, . , . , n), each bearing 
a charge g<, Discontinuities in the properties of the dielectric may be 
replaced by thin transition layers without affecting the generality of the 
theorem and we shall, therefore, assume the inductive capacity e of the 
medium to be a continuous but otherwise arbitrary function of position. 
It may also be assumed that there is a density of charge throughout the 
volume of the dielectric, although such a condition does not often occur 
in practice. 

At every point within the dielectric the field of the charges in equi- 
librium must satisfy the conditions : 

(55) y . D = p, v x E = 0, E = - V<; 
over the surface of each conductor >S 

(56) $< = constant, f D n da = #<; 

JSi 

at infinity the potential vanishes as 1/r. 1 

Suppose that <', E', D' is any other possible electrostatic field; it satis- 
fies the conditions (55), but not necessarily (56), and is known to differ 
somewhere, if not everywhere, from <, E, D. Since the volume distri- 
bution p and the total charge on the conductors is fixed we have 

(57) V (D' - D) = 0, I (D' - D) - n da = 0. 

Js* 

If U and U' are the electrostatic energies of the two fields, their difference 
is 

(58) U' - U = i J E' D' dv - i f E - D dv, 
or, since D' = cE', 

(59) W - U = I J (E' - E) . (D' - D) dv + f E (D' - D) dv. 

The second term on the right vanishes, for we see that on putting E = P, 
D' D = Q, the conditions of the theorem demonstrated in Sec. 2.9 
are satisfied. There remains 



(60) 



' ~ U = \ J * (E ' " 



which is an essentially positive quantity. The theorem is proved, for 
if E' differs in any region of space from E the resultant energy U' will be 
greater than U. The condition of electrostatic equilibrium is character- 
1 See Sec. 3.5, p. 167. 



116 STRESS AND ENERGY [CHAP. II 

ized by a minimum value of electrostatic energy and one therefore con- 
cludes that for the determination of equilibrium the energy U plays the 
same role in electrostatics as the potential energy in mechanics. 

2.12. Earnshaw's Theorem. A charged body placed in an electro- 
static field cannot be maintained in stable equilibrium under the influence of 
the electric forces alone. 

We shall suppose that the initial field is generated by a set of charges 
qi, (i = 1, . . . , n), distributed on n fixed conductors whose bounding 
surfaces are denoted by Si. These conductors are embedded in a dielec- 
tric whose inductive capacity may be a continuous function of position 
but in which there is no volume distribution of charge. We note first 
that neither the potential nor any of its partial derivatives can assume a 
maximum or a minimum value at a point within the dielectric; for if <t> 
is to be an absolute maximum it is necessary that the three partial deriva- 
tives 3 2 (t>/dx*, d 2 <t>/dy*, d*<t>/dz* shall all be negative at the point in ques- 
tion, but this condition is incompatible with Laplace's equation 



Likewise the existence of an absolute minimum requires that these three 
derivatives shall be positive, which again is inconsistent with (61). 
The same argument applies also to the derivatives of the potential. 

Let us suppose now that a charge qo is placed on a conducting surface 
So. The distributions on all the conductors are momentarily assumed to 
be fixed and S Q is introduced into the field of the other n charges. If co 
is the surface density of charge on /So, the energy of this conductor is 



(62) C/o = ^ 0o)o da, 

* JSo 

where < is the potential of the initial field. Let x, y, z be the coordinates 
of any point which is fixed with respect to So and , rj, f, those of any 
point on the surface. The potential on So due to the other n charges 
may be represented by a Taylor series in terms of its value and the value 
of its derivatives at z, t/, 2, 

(63) *(*, i,, r) = *(*, y,z) + - x) + (i, - y) + (T - z) 



hence the energy, too, may be referred to the potential and its derivatives 
at this point. But since < cannot be a minimum at x, y, z, it is always 
possible to displace the conductor So in such a way that the energy C/ 
is decreased. If after this displacement the charges, which thus far 
have been assumed to be "frozen" on the surfaces S , St, are released, 



SBC. 2.13} UNCHARGED CONDUCTORS 117 

these surfaces will again become equipotentials and by Thomson's 
theorem the energy of the field will be still further diminished. A 
minimum value of the energy function U Q (x, y, z) does not exist in the 
electrostatic field and consequently the conductor S Q is never in static 
equilibrium. 

2.13. Theorem on the Energy of Uncharged Conductors. The intro- 
duction of an uncharged conductor into the field of a fixed set of charges 
diminishes the total energy of the field. 

The conditions are the same as in the previous theorem but the 
surface SQ now carries no charge. Let E, D be the vectors characterizing 
the field before, E', D' the field after the introduction of the conductor /S . 
The change in energy is 



(64) 



U - W = ^ J E - D dv - \ f E' D' dv. 



The volume integrals are to be extended through all space, but since the 
field vanishes in the interior of the surfaces /S , /S, the contribution from 
these regions is zero. Let V be the volume of the field exterior to the n 
conductors Si before the introduction of $o, and VQ the volume bounded 
by /So. Then 

(65) V l = V - 7 

represents the volume of the region occupied by the field after the intro- 
duction of /So, and Eq. (64) may be written in the form 

(66) U - V = J f E D dv - ~ ( E'-V'dv 

* Jv 2 Jvi 

= J f E - D dv + i f (E D - B' D') (to 

* JVo * JVi 

= 5 f E D dv + \ f (E - BO (D - D') dv 

* JVo * JVi 

+ f E' (D - D') dv. 

JVi 

The last integral on the right can be shown to vanish. For 

E' (D - D ; ) = - v*' (D - DO = -V [<'(D - DO] 

+ 4/V-(D -D 7 ), 
and V (D DO = 0. By the divergence theorem, 



(67) 



f E' (D - DO dv = f V [0'(E>' - D)] dv 

JVi JVi 



118 STRESS AND ENERGY [CHAP. II 

where the <j are the potentials of the conducting surfaces Si. The 
total charge on each surface is constant, so that 

C f 

(68) I D' n da = I D n da = j<, 

/5* t/S 

and hence 

(69) I (D' - D) n da = 0. 

%/Si 

The difference between the initial and final energies is, therefore, 

(70) U - V 9 = I f *E*dv + ~ f e(# - ') 2 tfo, 

2 JFO 2 J Fl 

an essentially positive quantity. 

MAGNETOSTATIC ENERGY 

2.14. Magnetic Energy of Stationary Currents. Let us consider a 
stationary distribution of current confined to a finite region of space. 
This current may be supported by conducting matter, or result from the 
convection of charges in free space. The equation of continuity reduces 
to v J == 0, in virtue of which we may imagine the distribution to be 
resolved into current lines closing upon themselves. A current tube or 
filament may be constructed from the current lines passing through an 
infinitesimal element of area. The tube is bounded by those lines which 
pass through points on the contour of the element. At every point 
on the surface of a tube the flow is tangential; no current leaves the tube 

and consequently the net charge trans- 
ported across every cross section of the 
tube in a given time is the same. 

We shall calculate first the potential 
energy of a single, isolated current filament 
Sr ^n 3 in the field B of fixed external sources. 

FIG. 19. illustrating displacement The current carried by the filament is / 

of a current filament. 1-1 < 11 -A- i j j. J 

and it follows an initial contour designated 

in Fig. 19 as Ci. The force exerted by the field on a linear element of the 
filament is 

(1) f = I ds X B. 

Suppose now that the filament Ci is translated and deformed in such a 
manner that every element ds is displaced by the infinitesimal amount 6r 
into the contour C 2 . The displacement 6r is assumed to be a continuous 
function of position about C\ but is otherwise arbitrary. The work 
done by the force (1) in the displacement of the element ds is 

(2) f 8r = /(ds X B) dr = IB (5r X ds). 




SEC. 2.14] MAGNETIC ENERGY OF STATIONARY CURRENTS 119 

Let Si be any regular surface spanning the contour C\. A second 
surface S 2 is drawn to span C 2 , but in such a way as to pass through C\ 
and then coincide with Si The surface 

(3) &3 D 2 &l 

is, therefore, a band or ribbon of width 5r bounded by the curves Ci and 
C 2 . If da$ is an element of S 3 we have clearly 

(4) dr X ds = n 3 da 3 , 

where n 3 is the positive unit normal to $3. The positive faces of Si 
and 82 are determined by the usual convention that an observer, circu- 
lating about the contours in the direction of the current, shall have the 
positive face at his left. The magnetic fluxes threading Ci and C 2 are 

(5) $1 = f B m dai, $ 2 = f B n 2 da 2 ; 

t/oi /O2 

hence the net change in flux resulting from the displacement is 

(6) 5$ = $ 2 <J>i = f B n 3 da 3 . 

7*33 

On the other hand the total work done by the mechanical forces is 
obtained by integrating (2) around the closed contour Ci, whence in 
virtue of (4) we find 

(7) dW = < f Sr = I 5$. 

J Ci 

If now it be assumed that in the course of the virtual displacement 5r 
the external sources and the current / are maintained strictly constant, 
then the work done by the mechanical forces is compensated by a decrease 
in a potential energy function U. 

(8) SU = -dW = -7 8$, 

or 
(9) 



where S represents any surface spanning the current filament. Inversely, 
the mechanical forces and torques on a current filament in a magneto- 
static field can be determined from a variation of the function U while 
holding constant the current I and the strength of the sources. 

If 6r is a real rather than a virtual displacement, work must be 
expended to keep the current constant. The change 5$ in the flux 
induces an electromotive force 

(10) V = f B ds = - -^i 

JC n 



120 STRESS AND ENERGY [CHAP. II 

where dt is the time required to effect the displacement. This induced 
e.m.f. must be counterbalanced by an equal and opposite applied e.m.f. 
V. The work expended by the applied voltage on the circuit in the time 
dt is 

(11) VI U = +1 d$. 

The work done by the transverse mechanical forces in a small displacement 
of a linear circuit is exactly compensated by the energy expended by longi- 
tudinal electromotive forces necessary to maintain the current constant. 
The total work done on the circuit is zero. 

Let us suppose now that the source of the field B is a second current 
filament. Thus we imagine for the moment that the current distribution 
consists of just two isolated, closed filaments /i and I 2 . The fields 
generated by these currents are respectively BI and B 2 . The potential 
energy of circuit 7 2 in the external field BI is 

(12) f/ 2 i = -J 2 Bi-n 2 da 2 , 

JSt 

where S 2 is any surface spanning the filament 7 2 . Likewise the potential 
energy of circuit I\ in the field of I 2 is 



(13) f/i 2 = -Ii f B 2 

/01 



Uiz and f/2i are scalar functions of position and circuit configuration 
whose derivatives give the forces and torques exerted by one filament 
on the other. From the equality of action and reaction it follows that 

(14) Un = Vn = C/, 

and hence the mutual potential energy may be expressed as 

(15) U = -ili*i - i/A, 

where 3>i is the magnetic flux threading the filament I\. 

The mutual energy [/, which reduces to zero when the separation 
of the filaments becomes infinite, is not equal to the total work that must 
be done in approaching them from infinity to some finite mutual configura- 
tion, and therefore does not represent the total energy of the system. 
For let us suppose that 7 2 is allowed a small displacement under the 
forces exerted by Ji, with a resultant decrease dU in the potential 
energy. To maintain the current J 2 constant during the displacement, 
an equal amount of work dW must be done on the circuit J 2 to compensate 
the effect of the induced e.m.f. Thus far the net change in energy is zero. 
But now we must note that the displacement of 7 2 results in a change 
in the flux threading Ji and gives rise, therefore, to an induced e.m.f. V\ 
opposing the current /i. If I\ is to be maintained constant, a voltage V[ 



SBC. 2.14] MAGNETIC ENERGY OF STATIONARY CURRENTS 121 

must be impressed on the circuit I\ compensating Vi and doing work 
at the rate IiV(. The induced e.m.f., however, depends only on the 
relative motion of Ii and J 2 - The work done by the electromotive forces 
induced in circuit (1) by th<* displacement of circuit (2) must equal that 
which would be done were (2) fixed and an equal and opposite displace- 
ment imparted to (1). The work done on (1) is therefore 8W. In sum ; 
when the mechanical forces acting on the two filaments are allowed to 
do work, there is a decrease in the mutual potential energy of amount 
dUj but this is offset by the work 2dW which must be done on the 
system to maintain I\ and I 2 constant. If the total magnetic energy 
of the system be denoted by T 7 , the variation in T associated with a 
relative displacement at constant current is 

(16) dT = 2dW - dU = 5W = -5C7, 
or, by (15), 

(17) T = i/i*i + i/ 2 <i> 2 . 

In general, if the current system is composed of n distinct current fila- 
ments, the magnetic energy is 



where $< is the flux threading circuit i due to the other n 1 circuits. 

Equation (18) is an expression for the work necessary to bring n 
closed current filaments from an initial position at infinity to some speci- 
fied finite configuration. The cross section of any filament is very small 
but does not vanish, and a filament in the sense that we have used it here 
is not, therefore, a line singularity. Each filament or tube may be sub- 
divided into a bundle of thinner filaments carrying fractions of the initial 
current. In the limit of vanishing cross section, the current carried 
by the filament is infinitesimally small, as is also the energy necessary 
to establish it. Consequently the total energy of a current distribution 
is simply the mutual energy of the infinitesimal filaments into which the 
distribution can be resolved. 

Now in practice a current distribution in a conducting medium cannot 
be established by the mathematically simple expedient of collecting 
together current filaments from infinity, and although (18) proves to be 
correct for the magnetic energy of n circuits in a medium for which the 
relation of H to B is linear, this is not necessarily true in general. In 
place of a continuous distribution let us consider for the moment n 
linear circuits embedded in any magnetic material. The resistance of 
the ith circuit is JB< and at any instant the current it carries is /. To each 
circuit there is now applied an external e.m.f. 7< generated by chemical 



122 STRESS AND ENERGY [CHAP. II 

or mechanical means. These applied voltages give rise to variations 
in the currents and corresponding variations in the magnetic flux thread- 
ing each circuit. If Vi is the voltage induced by a variation in <f> t , the 
relation of current to total e.m.f. in the circuit at any instant is 

(19) y< + Vi = 

rr d$i 

or, since Vi = 



(20) V' { = 

UrV 

The power expended by the impressed voltage V^ is F<I, and conse- 
quently the work done on the ith circuit in the interval dt is 

(21) dWi = R>Pi dt + Ii d$i. 

Of this work the amount RJ\ dt is dissipated as heat, while the quantity 
Ii d$i is stored as magnetic energy. 1 A variation in the magnetic energy 
of the n filaments is therefore related to increments in the fluxes by 

n 

(22) dl 

and the total energy expended on the system, apart from that dissipated 
in ohmic heat loss within the conductors, as the currents are slowly 
increased from zero to their final values, is 

(23) r-;| /,<. 

$to is the flux threading the ith circuit at the initial instant when all 
currents are zero. This initial flux will be zero if there is no remnant 
magnetization of the medium about the circuits. Equation (23) can 
be interpreted as available energy stored in the magnetic field only when 
the relation of Ii to $1 is single-valued (no hysteresis), and (23) reduces 
to (18) only when the relation of H to B and consequently of Ii to $ is 
linear, such that Ii d<bi = <$; dli. 

It is a simple matter to extend (23) from a finite number of current 
filaments to a continuous distribution of current. Within this dis- 
tribution let us choose arbitrarily a surface element d<r. The current 
lines passing through points of da- constitute a current tube, which, in 
virtue of the stationary character of the distribution, closes upon itself. 
If J is the current density, the scalar product J n da = dl is constant 

1 A portion of the energy /< 53> t may, however, be made unavailable because of 
hysteresis effects. See Sec. 2.16. 



SKO. 2.15] MAGNETIC ENERGY AND FIELD INTENSITY 123 

over every cross section of the tube. Let B' be the magnetic field at any 
point within the tube produced by all those current filaments which lie 
outside. The increment in the energy of the current dl due to an 
infinitesimal increase in the field of the external filaments is 

(24) J(dT) = dl 64?' = dl f 5B' n da, 

the integral to be extended over any surface bounded by the contour of 
the filament dL If A' is the vector potential of the field B', then 
B' = V X A' and (24) may be transformed by Stokes' theorem to a line 
integral following the closed contour of the filament. 

(25) S(dT) = J n da SA' ds. 



Now the total vector potential A along any central line of the tube dl is 
equal to A' plus the contribution of the current dl = J n da itself ; but 
as the cross-sectional area da + 0, the latter contribution vanishes, so 
that in the limit A' may be replaced by A. Since furthermore the cur- 
rent density vector J, the unit vector n normal to da, and the element 
of length ds along the tube are parallel to one another within a tube of 
infinitesimal cross section, one may write in place of (25) : 

(26) o(dT) = (fj-B&dads. 

The product da ds = dv represents the volume of an infinitesimal length 
of tube. The total increment in the energy of the distribution is to be 
found by summing the contributions of all the tubes into which it has 
been resolved, and this summation is clearly equivalent to an integration 
of the product J 5A over the entire volume occupied by current. 

(27) 5 

The work required to set up a continuous current distribution by means 
of applied electromotive forces is, therefore, in general 

(28) r 

In case the relation between the current and the vector potential which 
it produces is linear, this reduces to 

(29) T 

2.16. Magnetic Energy as a Function of Field Intensity. We shall 
suppose that discontinuities in the magnetic properties of matter in the 



124 STRESS AND ENERGY [CHAP. II 

field can be represented by layers of rapid but continuous transition. 
The field may be due in part to currents, in part to permanent magnets 
or residually magnetized matter, but all sources are located within a 
finite radius of some arbitrary origin. It will be demonstrated in Chap. 
IV that under these circumstances the vectors A and B vanish at infinity 
as r~ l and r~ 2 . The current density J at any point is related to the 
vector H at the same point by 

(30) J = V X H. 
Furthermore, by a well-known identity, 

(31) J 6A = SA - V X H = V (H X 8A) + H - V X 6A 

= V (H X A) + H SB. 

Upon introducing (31) into (28) and applying the divergence theorem, 
we obtain 

(32) T = f f B H dB dv + f f A (H X dA) - nda, 

jv /BO / t/Ao 

where V is any volume bounded by a surface S enclosing all the sources 
of the field. If the surface S is allowed to recede toward infinity, the 
second integral of (32) vanishes, for the integrand diminishes as r~ 3 
whereas S grows only as r 2 . Therefore the work done by impressed 
e.m.fs. (such as might be derived from batteries or generators) in building 
up a magnetic field from the initial value B to the final value B can be 
represented by the integral 

(33) T 

extended over all space. It is to be emphasized that (33) is the energy 
associated with the establishment of a current distribution in the presence 
of magnetic materials, and does not include the internal energy of per- 
manent magnets or the mutual energy of systems of permanent magnets. 
The magnetic properties of all materials exclusive of the ferromagnetic 
group differ but slightly from those of free space; the relation between B 
and H is linear within wide limits of field intensity, the factor /i in the 
relation B = /zH is very nearly equal to MO, and there is no appreciable 
remnant magnetism. Under these circumstances the work done in 
building up the field to a value B is returned as the field is again decreased 
to zero. Equation (33) integrates to 

(34) 

which we interpret as the energy stored in the magnetic field. As in the 
corresponding electrostatic case we may suppose this energy to be dis- 
tributed throughout the field with a density |/x# 2 joules/meter. 3 



SEC. 2.16] 



FERROMAGNETIC MATERIALS 



125 



2.16. Ferromagnetic Materials. The relation of H to B for ferro- 
magnetic substances is in general nonlinear and multivalued. In an 
initial, unmagnetized state the vectors B and H are zero. If the field is 
now built up slowly by means of impressed electromotive forces applied 
to conducting circuits, the function B = B(H) at any point in the ferro- 
2.00 




200 400 600 800 1000 1200 1400 
H in ampere-turns per meter (rationalized) 
FIG. 20. Typical magnetization curve for annealed sheet steel. 

magnetic material follows a curve of the form indicated by Fig. 20. 
According to (33) the work done in magnetizing a unit volume of the 
substance is represented by the shaded area in the figure. Were the 
function B(H) single- valued, a decrease in the field from Hi, BI to zero 
would follow the same curve and the entire energy (33) would be avail- 
able for useful work. Actually, the return usually follows a path such 
as that indicated in Fig. 21. Starting 
at Hi, the field is decreased until 
H = 0. The associated value of 
B = B 2 , however, is still positive. 
To reduce B to zero, negative values 
must be imparted to H, meaning 
physically thai: H must be increased 
in the opposite direction. At 
H = H 3 the vector B is zero, and 
as H continues to increase in negative 
value a point is eventually reached 
where simultaneously H = Hi, 




FIG. 21. Hysteresis loop. 



B = BI. The return to the positive values HI, BI follows the sym- 
metrical path through B = -B 2 , H = and B = 0, H = +H 3 . At all 
points along the segment BiB 2 the value of B is greater than its initial 
value on the dotted curve for the same value of H ; the change of B lags 
behind that of H and the substance is said to exhibit hysteresis. 



126 STRESS AND ENERGY [CuAf. II 

Let w be the work done per unit volume of magnetic material in chang- 
ing the field from the value BI to B. 

(35) , 

If the variation of the field is carried through a complete cycle following 
the hysteresis loop from BI through B 2 , BI, B 2 and returning to BI, 
the net work done per unit volume is 

(36) w = -<j>B-dH, 

a quantity evidently represented by the enclosed area of the hysteresis 
loop illustrated in Fig. 21. The net work done per cycle throughout the 
entire field is 




(37) Q = -^ 

Q is the hysteresis loss, an irretrievable fraction of the field energy dis- 
sipated in heat. 

2.17. Energy of a Magnetic Body in a Magnetostatic Field. Let us 
suppose that a magnetic field BI from fixed sources has been established 

in a magnetic medium. We shall 
assume that the relation of BI to Hi 
is linear and that the medium is isotropic. 
Then BI = juiHi, where /n is at most a 
scalar function of position which reduces 
to a constant in case the medium is 
homogeneous. The energy of the field 
is 

FIG. 22. The region V\ is occupied * 

by a magnetic body embedded in the /OCA rp I TT . *D x/ 

homogeneous, isotropic medium Vz. ww * 1 2 I * * * tt 

extended over all space. Now reduce the intensity of the sources to 
zero and introduce into a suitable cavity formed in the first medium a 
body which we shall assume to be unmagnetized but whose magnetic 
properties are otherwise arbitrary. As in Sec. 2.10 the volume occupied 
by the embedded body will be denoted by Vi, Fig. 22, and the entire 
region exterior to it by F 2 - If the external sources are currents, the 
work which must be done in order to restore them to their initial intensity 
is 

(39) T 2 = f dv \ H-dB = ~ \ H* 1 Bdv+ f dv \ H-dB. 
JFI+FI Jo 2Jr, Jvi Jo 

The ultimate field B differs at every point from the initial field BI by 
an amount B 2 = B BI arising from the change in polarization of the 



SEC. 2.17] ENERGY OF A MAGNETIC BODY 127 

matter contained within V\. The increase in the work necessary to build 
up the current strength to the initial value is 

(40) T = T 2 - Ti = i f (H B - Hx Bi) do 

** \J r I 



It has been assumed that throughout the region 7 2 exterior to the body 
the relations between B and H are linear: 

(41) B! = Ml Hi, B = M1 H in 7 2 . 
The first integral in (40) is therefore equivalent to 

(42) \ I (H B - H ! BO dv = \ f (H - HO - (B + BO do. 

" JV* A J Vl 

By hypothesis the initial and final current distributions in the source are 
identical, so that in both Fi and F 2 the conditions 

(43) V - (B + BO - 0, V X (H - HO = 0, 

over the surface bounding V\ we 

, N n-[(B + BO+- (B + BO-] = 0, 

(44) 

n X [(H - HO+ - (H - HO-] = 0. 
The theorem of Sec. 2.9 may be applied, giving 

1 f 1 f 

(45) ^ (H - HO (B + BO dv = - (H - HO (B + BO dv 



are satisfied, while over the surface bounding Fi we have according to 
(18), page 37, 



(H - HO (B + BO dv = 0, 

Vl 

or 

(46) if (H - Hi) - (B + BO dv 

* JVz 

- -s I (H - HO (B + BO dv. 

* JVi 

The additional work required to build up the currents in the presence of 
the body can thus be expressed in terms of integrals extended over the 
volume occupied by the body. Upon introducing (46) into (40), we 
obtain 

(47) r - f (HI-B - H-B! -H-B + 2 (* n.dB\dv. 



128 STRESS AND ENEMY [CHAP. II 

If hysteresis effects are negligible (H is a single-valued function of B), 
the quantity T may be interpreted as the energy of the body in the mag- 
netic field of any system of constant sources; and the variation of T 
resulting from a virtual displacement of the body determines the mechan- 
ical forces exerted upon it. If furthermore the magnetic properties of 
the material within V\ can be characterized by a permeability /* 2 , so that 
B = /* 2 H, then (47) reduces to 



(48) 



T = I ( (Hi B - H BO dv = I f G* 2 - /*i)H H! do. 

* JVi *JVi 



The body was assumed to be initially free of residual magnetism; con- 
sequently the vectors B and H within V\ are related to the induced mag- 
netic polarization M by Sec. 1.6, 

(49) B = M o(H + M), 
and, when B = ju 2 H, this leads to 

(50) M = 

If, therefore, the cavity V\ was initially free of magnetic matter we may 
put /m = fi and write (48) in the form 



(51) T = ^ M B! dv, 

" JV\ 

an expression which corresponds to (51), page 113, for the electrical case 
in all but algebraic sign. This distinction, however, is fundamental. 
Whereas in the electrostatic case the work done by the external forces in a 
virtual displacement is accompanied by a decrease of the potential 
energy U, we shall learn soon that the mechanical forces exerted on a 
magnetized body are to be determined from the increase in T, so that T 
in this sense behaves as a kinetic rather than a potential energy. 

A useful analogue of Eq, (52), page 114, may be deduced for the 
magnetostatic field from (48). Suppose again that BI is the field of 
fixed sources either currents or permanent magnets and that at every 
point in space the permeability is specified by ^ = ju(z, ?/, z), a continuous 
function of position. If now the permeability is varied by an infinitesi- 
mal amount 5ju, the consequent change in the magnetic energy is 

(52) 8T = ~ J b,H*dv\ 

therefore, the magnetic energy of a distribution of matter in a fixed field is 

(53) T I J dv H* d/i, (M = Mo* m ) 



SBC. 2.18] POTENTIAL ENERGY OF A PERMANENT MAGNET 129 

provided the matter is initially unmagnetized and /* is independent of H. 
For paramagnetic material /x > JJL O , K m > 1, but in the case of diamag- 
netic matte . < /* and the magnetic energy of the initial field is dimin- 
ished by the introduction of diamagnetic bodies. 

2.18. Potential Energy of a Permanent Magnet. A rigorous analysis 
of the energy of material bodies in a magnetic field becomes very much 
more difficult when these bodies are permanently magnetized. In that 
case there is a residual field, which we shall denote by B , H , associated 
with the magnet even in the absence of external sources. The relation of 
BO to HO may still be expressed in the form 

(54) Bo = Mo(H + Mo), 

but Mo, the intensity of magnetization, is now quite independent of H , 
being determined solely by the previous history of the specimen. If an 
external field is applied, there will be induced an additional component 
of magnetization which, as in the past, we denote by M. This induced 
polarization M depends primarily on the resultant field H within the 
magnet (vanishing as H reduces to H ), but also on the state of the mag- 
net and its permanent or residual magnetization M . 

(55) B = M o[H + M(H, Mo) + Mo]. 

The resultant field H at an interior point is determined furthermore by 
the shape of the magnet as well as by the intensity and distribution of the 
external sources. 

We shall content ourselves with the derivation of a simple and fre- 
quently used expression for the potential energy of a system of perma- 
nent magnets. The proof rests on assumptions which are approximately 
fulfilled in practice; namely, that the magnetization M is absolutely 
rigid and that the induced magnetization M in one magnet arising from 
the external field of another is of negligible intensity with respect to M . 
The field at all points, both inside and outside a magnetized body occupy- 
ing a volume Vi, is exactly that which would be produced by a stationary 
volume distribution of current throughout Vi of density 

(56) J = V X Mo, 

together with a current distribution on the surface $ bounding Vi of 
density 1 

(57) K = Mo X n, 

where n is the unit outward normal to S. As a direct consequence of the 
analysis of Sec. 2.14 and in particular of Eq. (9), it follows that the poten- 
tial energy of the magnet V i in the field of other permanent magnets or of 
1 See Sees. 1.6 and 4.10. 



130 STRESS AND ENERGY [CHAP. II 

constant currents is 



(58) C/= 

or, in virtue of (56) and (57), 

(59) U = - f v (V X Mo) A dv - f s (M X n) A da. 

Now 

(60) A V X Mo = V (M X A) + M - V X A, 

(61) (M X A) -n = (n X M ) -A = -(M X n) - A, 

so that upon applying the divergence theorem and putting B = V X A, 
(59) reduces to 

(62) U = ~ f Vi Mo B dv. 

In (62) one may consider dm = M dv to be the magnetic moment of 
an element dv of the magnet. Its potential energy in the resultant field 
B is dU = B dm. The resultant field B is composed of the initial 
field Bi of all external sources, plus the field B 2 due to all other elements 
of the same magnet Vi. Therefore the work necessary to construct the 
magnet by collecting together permanently magnetized elements in the 
absence of an external field should be 

(63) t/2 = -f Vi Mo B 2 dv. 

These elements must be held together by forces of a nonmagnetic char- 
acter. The work necessary to introduce the magnet as a rigid whole 
from infinity to a point within the external field BI is then 



(64) Ui = - r Mo Bi dv, 

and the force exerted on a unit volume of the magnet by the external 
sources is 

(65) f = +V(M Bi). 

The difference of (64) and (51) is accounted for when we note that 
(64) is only the potential energy of the magnetized body in the external 
field, while (51) includes the work involved in building up the magnetiza- 
tion from zero to M, and is based on the assumption that there exists a 
linear relation between BI and Hi, and consequently between BI and M. 



SBC. 2.19] POYNTIN&S THEOREM 131 

ENERGY FLOW 

2.19. Poynting's Theorem. In the preceding sections of this chapter 
it has been shown how the work done in bringing about small variations in 
the intensity or distribution of charge and current sources may be 
expressed in terms of integrals of the field vectors extended over all space. 
The form of these integrals suggests, but does not prove, the hypothesis 
that electric and magnetic energies are distributed throughout the field 
with volume densities respectively 

(1) u = D E dD, w = B H - dB. 



The derivation of these results was based on the assumption of reversible 
changes; the building up of the field was assumed to take place so slowly 
that it might be represented by a succession of stationary states. It is 
essential that we determine now whether or not such expressions for the 
energy density remain valid when the fields are varying at an arbitrary 
rate. It is apparent, furthermore, that if our hypothesis of an energy 
distribution throughout the field is at all tenable, a change of field inten- 
sity and energy density must be associated with a flow of energy from 
or toward the source. 

A relation between the rate of change of the energy stored in the field 
and the energy flow can be deduced as a general integral of the field 
equations. 

(I) V X E + ^ = 0, (III) V - B = 0, 
(II) VXH- =J , 



We note that E J has the dimensions of power expended per unit 
volume (watts per cubic meter) and this suggests scalar multiplication of 
(II) by E. 

(2) E - V X H - E - = E . J. 

ot 

In order that each term in (I) may have the dimensions of energy pe* 
unit volume per unit time it must be multiplied by H. 

(3) H V X E + H 5 = 0. 

ot 

Upon subtracting (2) from (3) and applying the identity 
'4) V (E X H) = H V X E - E - V X H, 



132 STRESS AND ENERGY [CHAP. II 

we obtain 

(5) V (E X H) + E J = -E ^5 _ H ^ 

Finally, let us integrate (5) over a volume V bounded by a surface S. 

(6) f (E X H) n da + f E J dv = - f ( E ? + H ^ J do. 
JS Jv Jv \ ot dt/ 

This result was first derived by Poynting in 1884, and again in the 
same year by Heaviside. Its customary interpretation is as follows. 
We assume that the formal expressions for densities of energy stored in 
the electromagnetic field are the same as in the stationary regime. Then 
the right-hand side of (6) represents the rate of decrease of electric and 
magnetic energy stored within the volume. The loss of available stored 
energy must be accounted for by the terms on the left-hand side of (6). 
Let a be the conductivity of the medium and E' the intensity of impressed 
electromotive forces such as arise in a region of chemical activity the 
interior of a battery, for example. Then 

(7) J = <r(E + E'), E = J - E', 

a 

and hence 

f f ij 2 - f ' 

Jv * "~ Jvff Jv 

The first term on the right of (8) represents the power dissipated in 
Joule heat an irreversible transformation. The second term expresses 
the power expended by the flow of charge against the impressed forces, 
the negative sign indicating that these impressed forces are doing work 
on the system, offsetting in part the Joule loss and tending to increase 
the energy stored in the field. If, finally, all material bodies in the field 
are absolutely rigid, thereby excluding possible transformations of 
electromagnetic energy into elastic energy of a stressed medium, the 
balance can be maintained only by a flow of electromagnetic energy 
across the surface bounding V. This, according to Poynting, is the 
significance of the surface integral in (6). The diminution of electro- 
magnetic energy stored in V is partly accounted for by the Joule heat 
loss, partly compensated by energy introduced through impressed forces ; 
the remainder flows outward across the bounding surface S, representing 
a ?oss measured in joules per second, or watts, by the integral 

(9) f S n da = f (E X H) n da. 
js js 

The Poynting vector S defined by 

(10) S = E X H watts/meter 2 , 



SBC. 2.19] POYNTING'S THEOREM 133 

may be interpreted as the intensity of energy flow at a point in the field; 
i.e., the energy per second crossing a unit area whose normal is oriented 
in the direction of the vector E X H. 

It has been tacitly assumed that the medium is free of hysteresis 
effects. In case the relation between B and H is multivalued, energy to 
the amount Q [Eq. (37), page 126] is dissipated in the medium in the 
course of every complete cycle of the hysteresis loop. If the field is 
harmonic in time with a frequency v, there will be v cycles/sec, and con- 
sequently the hysteresis will also participate in the diminution of mag- 
netic energy at the rate of i>Q joules/sec. 

In the absence of ferromagnetic materials the relations between the 
field intensities are usually linear, and if the media are also isotropic 
Poynting's theorem in its differential form reduces to 

cr c 

As a general integral of the field equations, the validity of Poynting's 
theorem is unimpeachable. Its physical interpretation, however, is open 
to some criticism. The remark has already been made that from a 
volume integral representing the total energy of a field no rigorous con- 
clusion can be drawn with regard to its distribution. The energy of the 
electrostatic field was first expressed as the sum of two volume integrals. 
Of these one was transformed by the divergence theorem into a surface 
integral which was made to vanish by allowing the surface to recede to 
the farther limits of the field. Inversely, the divergence of any vector 
function vanishing properly at infinity may be added to the conven- 
tional expression u = E D for the density of electrostatic energy with- 
out affecting its total value. A similar indefiniteness appears in the 
magnetostatic case. 

A question may also be raised as to the propriety of assuming that 

vTV f$V\ 

E and H represent the rate of change of energy density for 

rapid as well as quasi-static changes. Such an assumption seems plaus- 
ible, but we must note in passing that a transformation of the energy 
function expressed in terms of the field vectors to an expression in terms 
of the densities of charge and current leads now to difficulties with surface 
integrals. Let 



be the rate at which work is done on the system by external forces. Upon 
introducing the potentials 



134 STRESS AND ENERGY [CHAP. H 

(13) E = -V<t> -^, B = V X A, 

of 

and applying the identities 

(14) T *'f =T 
H -^ = * 

we obtain 



where $ is a surface enclosing the entire electromagnetic field. Now in 
the stationary or quasi-stationary state the potentials can be shown to 
vanish at infinity as r~ l and the fields as r~ 2 ; the integral over an infinite 
surface then vanishes. It will be shown in due course, however, that 
the fields of variable sources vanish only as r" 1 and in this case the last 
term of (15) cannot be discarded by the simple expedient of extending the 
integral over an infinitely remote surface. On the other hand we know 
that fields and potentials are propagated with a finite velocity. If, 
therefore, the field was first established at some finite instant of the past, 
a surface may be imagined whose elements are so distant from the source 
that the field has not yet arrived. The intensity over S is then strictly 
zero and under these circumstances 

dW C f , dp , . dA\ 

J av. 



Finally, it must be granted that even though the total flow of energy 
through a closed surface may be represented correctly by (9), one can- 
not conclude definitely that the intensity of energy flow at a point is 
S = E X H; for there might be added to this quantity any vector 
integrating to zero over a closed surface without affecting the total flow. 

The classical interpretation of Poynting's theorem appears to rest to 
a considerable degree on hypothesis. Various alternative forms of the 
theorem have been offered from time to time, 1 but none of these has the 
advantage of greater plausibility or greater simplicity to recommend it, 
and it is significant that thus far no other interpretation has contributed 
anything of value to the theory. The hypothesis of an energy density 

1 MACDONALD, " Electric Waves," Cambridge University Press, 1902. LIVENS, 
''The Theory of Electricity," Cambridge University Press, pp. 238^., 1926. MASON 
and WEAVEB, "The Electromagnetic Field," University of Chicago Press, pp. 264 #., 
1929. 



SEC. 2.20] THE COMPLEX POYNTINO VECTOR 135 

in the electromagnetic field and a flow of intensity S = E X H has, on 
the other hand, proved extraordinarily fruitful. A theory is not an 
absolute truth but a self-consistent analytical formulation of the relations 
governing a group of natural phenomena. By this standard there is 
every reason to retain the Poynting-Heaviside viewpoint until a clash 
with new experimental evidence shall call for its revision. 

2.20. The Complex Poynting Vector. If we now let h = u + w 
represent the density of electromagnetic energy at any instant and 

Q = _ J2 _ j/ , j -faQ power expended per unit volume in thermo- 

G 

chemical activity, the Poynting theorem for a field free from hysteresis 
effects may be written 

(17) V-S + - + Q-0. 

In a stationary field h is independent of the time, so that (17) reduces to 

(18) V S + Q = 0. 

Q may be positive or negative as the work done by the impressed elec- 
tromotive forces E' is less or greater than the energy dissipated in heat. 
Accordingly the energy flows from or toward a volume element depending 
on its action as an energy source or sink. 

The sources and their fields in most practical applications of electro- 
magnetic theory are periodic functions of the time. The mean value of 
the energy density h is constant and dh/dt = dh/dt = 0, the bar indi- 
cating a mean value obtained by averaging over a period. In the case of 
periodic fields, therefore, 

(19) TTS + Q-0, or J^,S nda + f y Qdv = 0. 

When there are no sources within V, the energy dissipated in heat 
throughout V is equal to the mean value of the inward flow across the 
surface S. 

The advantages of complex quantities for the treatment of periodic 
states are too well known to be in need of detailed exposition. The 
reader may be reminded, however, that certain precautions must be 
observed when dealing with products and squares. Throughout the 
remainder of this book we shall usually represent a harmonic variation 
in time as a complex function of the coordinates multiplied by the factor 
e -iw B Thus if A is the quantity in question, we write 

(20) A = A 06-' = (a + ift)<r** = (a + i/3)(cos co - i sin 0, 

where a and ft are real functions of the coordinates x, y, z. The conjugate 
of A is obtained by replacing i = \/"~l by ~~i> & n d is indicated by the 



136 STRESS AND ENERGY [CHAP. II 

sign A. 

(21) 1 = (a - ifS)e^. 

Although it is convenient to employ complex quantities in the course of 
analytical operations, physical entities must finally be represented by 
real functions. If A satisfies a linear equation with real coefficients, 
then both its real and its imaginary parts are also solutions and either 
may be chosen at the conclusion of the calculation to represent the 
physical state. However, in the case of squares and products, we must 
first take the real parts of the factors and then multiply, for the product 
of the real parts of two complex quantities is not equal to the real part of 
their product. We shall indicate the real part of A by Re(A) and the 
imaginary part by Im(A). 

Re(A) = a cos co + ft sin orf = Va 2 + ft 2 sin (orf + <), 

(22) Im(A) = ft cos ut a sin ut = V 2 + ft 2 cos (co -f </>), 

< = tan- 1 - 

The square of the amplitude or magnitude of A is obtained by multi- 
plication with its conjugate. 

(23) AA = a 2 + ft 2 . 
The real part of A is also given by 

(24) 



The product of the real parts of two complex quantities A\ and A 2 is, 
therefore, equal to 

(25) Re(A 1 ) - Re(A*) = ^(A l + li)(A 2 + I,) 



The time average of a periodic function A is defined by 

- 1 f T 

(26) A = - I A dt, 

where T is the period. If A is a simple harmonic function of the 

its mean value is of course zero. The mean value of such functions as 

cos ut sin ut also vanishes, and we have consequently from (25) the result 



(27) Re(Ai) 

or 

(28) 



SEC. 2.21] BODY FORCES IN FLUIDS 137 

According to the preceding formulas the mean intensity of the energy 
flow in a harmonic electromagnetic field is 



(29) S = Re(E) X Re(S) == $Re(E X H), 

the real part of the complex vector E X H. The properties of this 
so-called complex Poynting vector are interesting. We shall denote it by 

(30) S* = IE X H. 

Let us suppose that the medium is defined by the constants 6, /*, v and 
that the field contains the time only in the factor e***'. Then the 
Maxwell equations in regions free of impressed electromotive forces E' are 

(31) V X E = tw/iH, V X H = (<r - icoe)E. 
The conjugate of the second equation is 

(32) V X H = (o- + ico )E, 
which when united with the first as in Sec. 2.19 leads to 

(33) V S* = -friB E + to ( H - H - ~ E - E\ 

\ * / 

or in virtue of (28) and (23), 

(34) V S* = -Q + teu(w - u). 

The divergence of the real part of S* determines the energy dissipated in 
heat per unit volume per second, whereas the divergence of the imaginary 
part is equal to 2w times the difference of the mean values of magnetic and 
electric densities. Throughout any region of the field bounded by a 
surface S we have 

(35) Re I S* n da = total energy dissipated, 

js 

and 

(36) Im I S* n da = 2w X difference of the mean values of magnetic 

js 

and electric energies. 
FORCES ON A DIELECTRIC IN AN ELECTROSTATIC FIELD 

2.21. Body Forces in Fluids. Let us consider an electrostatic field 
arising from charges located on the surfaces of conductors embedded in 
an isotropic dielectric. To simplify the analysis we shall assume for the 
moment that throughout the entire field the dielectric medium has no 
discontinuities other than those occurring at the surfaces of the con- 
ductors. It will be assumed furthermore that D = cE, where c is a 



138 STRESS AND ENERGY [CHAP. II 

continuous function of position. The total electrostatic energy of the 
field is therefore 

(1) U = f eE* dv 

extended over all space. 

The position of any point in the dielectric with respect to a fixed 
reference origin is specified by the vector r. Every point of the dielectric 
is now subjected to an arbitrary, infinitesimal displacement s(r). It is 
supposed, however, that the conductors are held rigid and the displace- 
ment of dielectric particles in the neighborhood of the conducting surfaces 
is necessarily tangential. The displacement or strain results in a varia- 
tion in the parameter c with a corresponding change in the electrostatic 
energy equal to 



(2) SU = ~ I deE*dv. 

At the same time a slight readjustment of the charge distribution will 
occur over the surfaces of the conductors. Initially the surface charges 
were in static equilibrium and the initial energy (1), according to Thom- 
son's theorem, Sec. 2.11, is a minimum with respect to infinitesimal varia- 
tions in charge distribution. The variation in energy associated with 
the redistribution is, therefore, an infinitesimal of second order which 
may be neglected with respect to (2). 

Consider the dielectric material contained initially in a volume dv^ 
The displacement is accompanied by a deformation, so that after the 
strain this element of matter may occupy a volume Eq. (54), page 92. 

(3) dv 2 = (1 + V s) dvi. 

The mass of the element is conserved and hence if the density of matter 
is denoted by r we have 

(4) TI dvi = T 2 (l + V s) dvi, 
or, for an infinitesimal change, 

(5) 5r = T2 TI = TV s. 

The inductive capacity e is a function of position in the dielectric and 
also of the density r. The element which after the displacement finds 
itself at the fixed point r was located before the displacement at the 
point r s, and the contribution to 5e arising from the inhomogeneity 
of the dielectric is therefore s Ve. If it be assumed that e depends 
only on r and r, the total variation is 

(6) de = S Ve + ~ 5r = -s Ve r ~ V s. 

or or 



SEC. 2.21] BODY FORCES IN FLUIDS 139 

The change in electrostatic energy associated with the deformation is 

(7) 817 = J \E*Ve s + E*r ~ V - sj do. 

Now 

(8) 

and hence 
(9) U. 

These volume integrals are to be extended over the entire field. The 
surfaces of the conductors, within which the field is zero, shall be denoted 
by Si, 82, . . . , S n . A surface So is drawn to enclose within it all 
conductors and all parts of the dielectric where the field is of appreciable 
intensity. Within the volume bounded externally by So and internally 

by /Si, 82, 5 S n the function E 2 r s is continuous. Thus the 
divergence theorem may be applied to the second integral of (9), giving 

(10) 

Over So the intensity of E is zero, while over the rigid conductors Si, 
. . . , S n the normal component of s has been assumed zero. The 
integral (10) therefore vanishes. 

If one neglects the effect of gravitational action, it may be assumed 
that the only body force f is that exerted by the field on elements of 
dielectric. The work done by this force per unit volume during the 
displacement is f s, and hence according to the principle of conservation 
of energy, 

(11) f f -scto = - f U# 2 Ve-i 

The displacement s is arbitrary and we find 
(12) 

In case there are also charges distributed throughout the dielectric, a 
term pE must be added. 

The last term of (12) is associated with the deformation of the dielec- 
tric. The assumption that e can be expressed as a function of position 
and density alone is admissible for liquids and gases but is not necessarily 



140 STRESS AND ENERGY [CHAP. II 

valid in the case of solids. To a very good approximation the relation 
of dielectric constant K, (e = Ke ) to density in gases, liquids, and even 
some solids is expressed by the Clausius-Mossotti law, 

/t n\ & """" 1 /T t 

( 13 ) r-iy = CT, or K - I = 



1-CV 

where C is a constant determined by the nature of the dielectric. Through 
a simple calculation this leads to 

(14) T^ = | (K-I)OC + 2). 

The force exerted by the field on a unit volume of a liquid or gas is, 
therefore, 



(15) f = - E*V* + V [E*(K - 1)(* + 2)]. 

2.22. Body Forces in Solids. The volume force (12) exerted on a 
dielectric by an electrostatic field has been derived on the assumption 
that the variation in the inductive capacity e during an infinitesimal 
displacement can be accounted for by the inhomogeneity of the dielectric 
and the change in density associated with the deformation. It is clear, 
however, that deformations in solids may occur without any accompany- 
ing change of volume, and that a rigorous theory must therefore express 
the variation of in terms of the components of strain. Since the 
dielectric is presumed to be in static equilibrium, the mechanical forces 
exerted by the field will be balanced by elastic forces induced during the 
deformation. Our problem is to find these forces and to determine the 
resultant deformation of the dielectric due to the applied field. We shall 
confine the investigation to media whose electric and elastic properties in 
the unstrained state are isotropic. Since the variations in e are dependent 
on the components of strain, it is hardly to be expected that a solid will 
remain electrically isotropic after the strain is applied. Our first task 
is to set up an expression for the electrostatic energy of an anisotropic 
dielectric. 

In Sec. 2.8 it was shown that the electrostatic energy density within 
a dielectric is 

(16) u = E dD 



without regard to the relation of D to E. We shall now assume that in 
an anisotropic medium the components of D are linear functions of the 
components of E. 

(17) J>*=S/*E, = 1.2,3). 



SBC. 2.22] BODY FORCES IN SOLIDS 141 

Since the quantities Ek, Dj transform like the components of vectors it is 
clear that the # are the components of a tensor of second rank. By a 
rotation of the coordinate system this tensor may be referred to principal 
axes defined by the unit vectors a, b, c such that 

(18) Da = a#a, A, = C b E b , D c = <5 # c . 

Then 

(19) E dD = e a E Q dE a + b E b dE b + * C E C dE c = D dE. 
The energy density in an anisotropic, linear medium is, therefore, 

(20) W = ^ 

We have next to calculate the change in the energy resulting from a 
variation of the parameters e# while holding the charges fixed. A review 
of the proof of Sec. 2.10 shows that it may be adapted directly to our 
present needs. To avoid confusion of subscripts we shall replace the 
index 1 in Sec. 2.10 by a prime to denote initial conditions. Thus the 
dielectric in the field was characterized initially by the coefficients cj fc . 
Within a region Fi these values are changed to ,*. The total change in 
the energy of the field, according to Eq. (49) , page 113, is 

(21) U = 
or in virtue of (17) 

(22) U = ~ [(ii ~ eiOEi-E'i + ( C22 - e' 22 )# 2 E 2 + (633 ~ 



+ (e 12 - e' n )(E 2 E( + EJEi) + ( 23 - ei 

+ (si - 
In case the variations in e,& are infinitesimal, this becomes 



(23) 



= -1 I 

* JVi 



Finally, the parameters ,-* must be related to the components of 
strain. For sufficiently small deformations we may put 

(24) &, 4 

The 81 coefficients a are the components of a tensor of rank four. In 
virtue of the relations c ? - fc = *,-, 6f m = e m i, we have 



142 STRESS AND ENERGY [CHAP. II 

Then of the 81 9 = 72 nondiagonal terms, only 36 are independent and 
the total number of independent coefficients is reduced to 36 + 9 = 45. 
To reduce the system still further use must be made of symmetry condi- 
tions and we now introduce the limitation to a medium which is initially 
isotropic, 1 though not necessarily homogeneous. 

The variation in the density of the electrostatic energy due to a small 
deformation is 

3333 

(26) su = -\ 2 2 2 2 a *> EiEh Seim - 



The coefficients a?f m have fixed values characteristic of the dielectric at 
each point, but varying from point to point in case the medium is inhomo- 
geneous. Now since the dielectric is assumed to be initially isotropic, 
Eq. (26) must be invariant to a reversal in the direction of any coordinate 
axis and to the interchange of any two coordinate axes. Thus a reversal 
of the axis x 3 - reverses the signs of E/ and e^ k ^ j, but leaves all other 
factors unchanged. As a consequence, certain coefficients must vanish 
if 8u is to be unaffected by the reversal or exchange of axes. In fact it is 
evident that all but three classes of coefficients are zero, namely: 

(27) <% = ai, a& = a 2 , a]l = a 3 , ( j ^ k). 
Equation (26) is thereby reduced to 

(28) du = -ifoxCEJ fen + El de, 2 + El 5e 33 ) 

+ aJi(El + ED den + (El + El) 5c 22 + (E\ + El) Se 33 ] 



Now 8u must also be invariant to a rotation of the coordinate axes 
and this implies a further relation between the parameters 01, a 2 , a 3 . 
The condition is easily found by rewriting (28) in the form 

1 f 3 3 

(29) du = (a x - a 2 - 4a 3 ) ^ E * be + a * E * 2 tey/ 

L j-i y=i 

+ 4a 3 (f 8e n + El de^ + El de^ + EiE 2 5ei 2 



3 

in which E* = El + El + El The sum ] e^ is, according to (54), 

; = 1 

page 92, the cubical dilatation, a quantity not dependent on the coordi- 
nate system. The middle term of (29) is, therefore, invariant to a rota- 
tion of the reference system. The same is true also of the last term; for 
if the strains e^ are replaced by the coefficients 2a/ fc of Sec. 2.2, this last 

1 The anisotropic case is discussed by POCKELS, Encyklopadie der inathematischon 
Wissenschaften, Vol. V, Part II, Teubner, 1906. 



SBC. 2.22] BODY FORCES IN SOLIDS 143 

term is an invariant quadric similar in form to the strain quadric of 
Eq. (36), page 89. Only the first term of (29) is variant with the coor- 
dinates and it is necessary that 1 

(30) ai a 2 4a 3 = 0, a 3 = (ai a 2 ). 

The variation in the density of electrostatic energy due to the local 
deformation, or pure strain, of an isotropic dielectric is finally 



(31) du 8 = -i[(ai#? + a 2 El + a 2 S 2 3 ) fen 



The subscript s has been added to emphasize the fact that this is only 
that portion of the total variation which arises from a pure strain. The 
most general deformation of an clastic medium, it will be remembered, 
is composed of a translation, a local rotation defined by (33), page 88, 
and the local straip. defined by (29), page 88, of which we have taken 
account in (31). In an anisotropic dielectric the local rotation gives 
rise to variations in the tensor components # and consequently a system 
of torques must act on each volume element. 2 In the initially isotropic 
medium considered here these rotational variations are absent. The 
variation in the density of electrostatic energy associated with an infini- 
tesimal translation 6s of an inhomogeneous dielectric was calculated in 
Sec. 2.21. 



(32) Su t = iE^Ve 6s, 

where e is the inductive capacity in the unstrained, isotropic state. 

A deformation of the dielectric occasions a variation in the elastic 
energy as well as in the electrostatic energy. According to (68), page 94, 

(33) 8u e = (XiV s + 2X 2 en) 5e n + (XiV s + 2X2622) 6e 22 

+ (XiV s + 2X2633) 



Now the work done by the mechanical forces acting on the dielectric 
within the volume Vi and over the surface Si bounding Vi during an 
infinitesimal displacement 5s must equal the decrease in the total available 
energy, elastic plus electrostatic. We write, therefore, for the energy 
balance: 

1 The procedure followed here in reducing the constants of electrostriction to two 
is identical with that which leads to the reduction of the elastic constants c,*, Sec. 2.3, 
to the two parameters Xi and \2. See, for example, LOVE, "Treatise on the Mathe- 
matical Theory of Elasticity," Chap. VI, 4th ed., Cambridge Press, 1927. The 
determination of these constants for various classes of anisotropic crystals was made 
by Voigt, "Kompendium der theoretischen Physik," Vol. I, pp. 143-144, Leipzig, 
1895. His results are cited by Love, loc. cit. 
2 POCKELS, loc. cit., p. 353. 



144 STRESS AND ENERGY [Ca^p. II 

(34) f f 5s dv + C t 5s da = -5 f (u. + u. + u t ) dv. 

J Vi JSi J Vi 

It will be convenient to resolve the total body force per unit volume f 
arbitrarily into the two components f and f", where 

(35) f = -p? 2 Ve 

is the contribution resulting from the inhomogeneity of the dielectric 
and where f " is the force associated with a pure strain. The latter must 
satisfy the relation 

(36) f f" 5s dv + f t" 6s da = -S f (u. + u.) dv. 

Jr\ ft/Ol J Y 1 

The resolution into translational and strain components of body and 
surface force is such that 

(37) f f dv + f t' da = 0, f f" cfo + f t" da = 0. 

/ rl /Ol / r 1 /Ol 

Then by Sec. 2.3, the left-hand side of (36) may be transformed to 



(38) 5W" = Y (T'A Sen + T& 5e 22 + T' 3 ' 3 de 33 + T 5e 12 + T& 8e M 

+ T' t { Sen) dv. 

The variations in the strain components are arbitrary and hence on 
equating coefficients of corresponding terms from (31) and (33) we obtain 

Til = -(XiV s + 2Xien) + 

ZM = -(XiV s + 2X 2 e 22 ) + 

TiS = -(XiV s + 2X 2 e, 3 ) + 
(39) 



These are the components of a stress tensor whose negative divergence 
gives us the resultant body force associated with a strain. If the body 
force is solely of electrical origin, as will be the case when gravitational 
action is neglected, the divergence of the elastic stresses will vanish and 
we obtain for the ^-component of force: 



(40) ft - - (oiE? + a 2 ^ + HBJ) - -~ ((a, - 



SBC. 2.22] BODY FORCES IN SOLIDS 145 

with analogous expressions for /' 2 ' and /". To these are added the com- 
ponents of f from (35) to obtain the total body force exerted by an 
electrostatic field in an isotropic dielectric. 

The parameters a\ and a 2 must in general be determined for a given 
substance by measurement. Physically the parameter a\ expresses 
the increment of corresponding to an elongation parallel to the lines of 
field intensity, while a* determines this increment for strains at right 
angles to these lines. 

If the dielectric within V\ is homogeneous and contains no charge, the 
gradient of is zero and the vector E satisfies V E = 0, V X U = 0. 
In this case f = and the resultant force I" = f reduces to 

(41) f = -i(ai + az)VE\ 

In a liquid or gas the shearing strains e ; -& are all zero and hence by 
Eqs. (24) and (27) a 3 = a$ = 0, ai = a 2 = a. Furthermore there can 
be no preferred directions in the dielectric properties of a fluid and con- 
sequently ,* = when j ^ k, en = e 22 = 33 = e. In place of (24) we 
may write: 

(42) 

^ ' 



den 
Denoting the cubical dilatation by A = e n + e^ + e 33 , we 



(43) 

Now by (5) 



dr de 



and consequently the total force per unit volume exerted by an electro- 
static field on a fluid dielectric is 

(45) 

as was deduced directly in Sec. 2.21. 

The derivation of the volume forces from an energy principle as in 
the foregoing paragraphs seems to have been proposed first by Korteweg 1 
and developed by Helmholtz 2 and others. A complete account of the 
theory with references to the older literature is given by Pockels. 3 The 
energy method has been criticized by Larmor 4 and Livens 5 who propose 

1 KORTEWEG, Wied. Ann., 9, 1880. 

2 HELMHOLTZ, Wied. Ann., 13, 385, 1881. 

8 POCKELS, Arch. Math. Phys., (2) 12, 57-95, 1893, and in the Encyklopadie der 
mathematischen Wissenschaften, Vol. V, Part II, pp. 350-392, 1906. 

LARMOR, Phil. Trans., A. 190, 280, 1897. 

8 LIVENS, Phil. Mag., 32, 162, 1916, and in his text ''The Theory of Electricity," 
p. 93, Cambridge University Press, 1926. 



146 STRESS AND ENERGY [CHAP. II 

alternative expressions for the forces. These criticisms, however, do not 
appear to be well founded. Objections to the particular form of the 
energy integral employed by Helmholtz are satisfied by a more careful 
procedure. Livens has undertaken a generalization of the energy method 
to media in which the relation between D and E is nonlinear in order to 
show that it leads to absurd results; in so doing he has omitted the essen- 
tial term associated with the deformation. The alternative expressions 
for the volume force proposed by Livens may on the other hand be derived 
very simply, in both the electric and magnetic cases, by assuming that 
polarized matter is equivalent to a region occupied by a charge of density 

p' = -v . P and current of density J' = + V X M (cf. Sec. 1.6). 

The forces may then be calculated exactly as in Sec. 2.5. Such a pro- 
cedure would be justifiable were the medium absolutely rigid. According 
to the Livens theory one calculates the force exerted by the field on the 
polarized matter, and then introduces these forces into the equations of 
elasticity to determine the deformation. The deformation, however, 
affects the polarizations and the two parts of the problem cannot, in 
general, be handled separately in any such fashion. Under certain 
conditions, particularly in fluids, the two theories lead to identical results, 
but in most circumstances they differ. There appears to be little reason 
to doubt that the energy method of Korteweg and Helmholtz is funda- 
mentally sound. 

2.23. The Stress Tensor. We shall suppose again that Si is a closed 
surface drawn within an isotropic dielectric under electrical stress. The 
properties of the dielectric are assumed to be continuous across this 
surface and at all interior points; it is not, in other words, a surface of 
discontinuity bounding a whole dielectric body. The total force exerted 
on the matter and charge within Si is 

(46) F = 

We wish to show now that the force F can be represented by a surface 
integral over Si. To this end it is only necessary to express the integrand 
as the divergence of a tensor. The term f" appears already in this form 
in (40) and therefore needs no further transformation. In the first term 
we replace p by V D and then make use of the identity 

(47) B 2 Vc = V(# 2 ) - 2(D V)E, 

which holds only when V X E = 0. It follows without difficulty that the 
components of the vector pE %E 2 Ve are 

3 

/AQ\ ^T? ^ J?2 ^S^ /I? n '\ 1 CF .,._- _^v 
1*OJ P*-J i ~"~" """ F J ' ' ^ \-*- J i'*^kJ """" C H i/) 



SBC. 2.24] SURFACES OF DISCONTINUITY 147 

and the integrand of (46) can thus be expressed as the divergence of a 
tensor 2 S, 

(49) p?-iS 2 Ve + f" = V- 2 S, 
whose components S# are tabulated below. 

Sn = ~ (E\ El - El) H (aiE\ + a 2 El + 

4 & 

e -w w JL 2 "" ai z? F Q 

012 ^1^2 T" o -^1^2 = >->21> 

(50) 2 ^ 



Upon applying the divergence theorem expressed in Eqs. (22) and 
(23), page 100, it appears that the force F exerted on the charge and 
dielectric matter within Si is equivalent to the integral over Si of a 
surface force of density 

(51) t = 



where n is as usual the outward unit normal to an element of Si. To 
maintain equilibrium an equal and opposite force per unit area must be 
exerted by the material outside Si on the particles composing Si. 

The tensor components (50) and the surface force t differ from the 
corresponding expressions developed for free space in Sec. 2.5 in that e 
is replaced by e and the deformation is accounted for by the constants ai 
and a 2 , thus giving rise to additional terms which may well be important. 
At any point on the surface Si the vector t lies in the plane of E and n, 
but is no longer oriented such that E bisects the angle between n and t 
as was the case illustrated by Fig. 17, page 102. 

2.24. Surfaces of Discontinuity. Throughout the previous analysis 
it has been assumed that abrupt discontinuities in the properties of the 
medium, such as must occur across surfaces bounding dielectric bodies, 
may be replaced by layers of rapid but continuous transition. As the 
transition layer becomes vanishmgly thin, a discontinuity is generated 
that gives rise to infinite values for the gradient of c. But the volume 
within the layer also vanishes and we must seek the limit of the force f 
per unit volume to learn what total force is exerted on the resultant 
surface of discontinuity. 

In Fig. (23) the crosshatched area represents a section of a transition 
layer between dielectric media (1) and (2). The force exerted by the 



148 



STRESS AND ENERGY 



[CHAP. II 



field on the matter and charge within this layer can be found by inte- 
grating (51) over the two faces. The two essential parameters that 
characterize an isotropic dielectric will be denoted as 



(52) 



a = c 



The force acting on unit area in the direction from medium (1) towards 
(2) as the thickness of the layer approaches zero is then 



(53) 



t = [E(E . n)] 2 + [aE(E n)h 



the subscripts indicating that the values are to be taken on either side 
of S. It should be noted that ni = n 2 . 

Very few reliable measurements of the constants ai and a 2 for solids 
have been reported in the literature. The traction t may, however, be 

computed in the case of an interface separat- 
| * ing two fluids to which the Clausius-Mossotti 

law can be applied. According to Eqs. (43) 



Dielectric (1) 



Dielectric (2) 



and (14), 

(54) a\ = a 2 = 
Then for a fluid, 

(55) <* = e, f 



~ 



1 



2). 



= - ( 2 - 2* - 2). 



At the interface the tangential components of 
^S l E are continuous, E n = EM, and in the 

FIG. 23. Section of a tran- absence of surface charge we have also 

sition layer separating two ^7 ^ rpi n ii-o' /ro\ xi, 

dielectric media. ^ E ni = 2 # n2 . The field E in (53) may, there- 

fore, be expressed in terms of its intensity in 

either medium. The values of a and )3 from (55) introduced into (53) 
lead to 



(56) t = ^ - 



?K 2 - 6*1*1 ~ 3/cf 



- 2) - 



The traction exerted by the field on the interface of two fluids is normal 
to the surface, directed from medium (1) toward (2), a necessary con- 
sequence, of course, of the fact that a fluid supports no shearing stress, 

In case medium (2) is air, K 2 may be placed approximately equal to 
unity. Equation (56) then reduces to 



(57) 



t = 



Q(KI - I) 2 

6 



SEC. 2.25] ELECTROSTRICTION 149 

The traction is a maximum when E is normal to the surface, and is in 
the direction of diminishing inductive capacity (cf. page 114). However, 
if E lies in the surface, E n i = 0, and the traction becomes negative, 
implying a pressure exerted by the field on dielectric jci. 

Another case of interest is that in which medium (1) is a conductor. 
Then Ei = 0, En = and (53) reduces to 

(58) t= ( a - p)E*n = -^ E*, 

all quantities having their values just outside the metal surface in the 
dielectric (1). In case this dielectric is a fluid, 



These expressions are valid whether or not the conductors are charged, 
for E n is related to the surface charge by eE n = co. But Eqs. (58) and 

(59) do not represent the resultant force per unit area exerted on the surface, 
for they will be compensated to a certain degree by stresses or pressures 
generated by the field within the dielectric. These we shall now investi- 
gate. Only in case medium (1) is free space or a gas of negligible induc- 
tive capacity can this internal pressure be ignored. We have then 

T = and (59) becomes 

(60) t = ie #*n = |coJ57 n n, 

where w is the charge per unit area. 

2.25. Electrostriction. The elastic deformation of a dielectric under 
the forces exerted by an electrostatic field is called electrostriction. The 
displacement s at any interior point of an isotropic substance must 
satisfy Eq. (70), page 95: 

(61) f + (Xi + X 2 )VV s + X 2 T 2 s = 0, 

where now the body force f = f'+ f" is given in general by (35) and 
(40). To these there may be added, when the occasion demands, a 
force Ep to account for a volume charge, and a gravitational force rg. 
Solutions of (61) must be found in an appropriate system of coordinates. 
These solutions are subject to boundary conditions over a surface bound- 
ing the dielectric. On this surface either the applied stresses or the 
displacement must be specified. Thus if the surface is free, stresses 
will be exerted upon it by the field and from these may be determined the 
boundary values of s. 

In solids the effect is small and easily masked by deformations arising 
from extraneous causes. The attractive forces between metal electrodes 
in contact with the dielectric may bring about deformations within the 



150 STRESS AND ENERGY [CHAP. II 

volume of the dielectric which have nothing to do with electrostriction. 
Experimental data on the subject are not abundant and frequently 
contradictory. The theory of electrostriction in a cylindrical condenser, 
however, has been developed in considerable detail. The method 
described above was applied by Adams, 1 and his results were later 
reconciled by Kemble 2 with the earlier work of Sacerdote. 3 An extensive 
list of references is given by Cady in the International Critical Tables. 4 
The electrostriction of fluids is more amenable to calculation, for the 
shearing modulus X 2 is then zero and Xi is equal to the reciprocal of 
the compressibility. According to (73), page 95, the pressure within the 
fluid is related to the volume dilatation by p = XiV s. This expres- 
sion, together with the body force f from Eq. (45), introduced into (61), 
leads to the relation 

(62) 

between pressure and field intensity at any point in the fluid. To inte- 
grate this equation the functional dependence of both pressure and 
inductive capacity on the density r must be specified. In a chemically 
nomogeneous liquid or gas, one may assume that p = p(r), e = c(r), 
and that the dependence on position is implicit in the independent variable 
r. Then 



(63) 

^ 

and hence (62) reduces to 



A scalar function P(p) of pressure is now defined such that 
(65) P(p) = JJ dp, VP = i Vp, 

where p Q is the pressure at a point in the fluid where E = 0. If ds is an 
element of length along any path connecting two points whose pressures 
are respectively p and p$, then VP ds = dP, and (64) is satisfied by 

(66) 

1 ADAMS, Phil. Mag., (6) 22, 889, 1911. 

8 KEMBLE, Phys. Rev., (2) 7, 614, 1916. 

* SACERDOTE, Jour, physique, (3) 8, 457, 1899; 10, 196, 1901. 

4 International Critical Tables, Vol. VI, p. 207, 1929. 



SBC. 2.26] FORCE ON A BODY IMMERSED IN A FLUID 151 

In liquids dp = Xi , Eq. (77), page 96, or 



p-po 

(67) T = T e x ' . 



The parameter Xi, the reciprocal of the compressibility, is a very large 
quantity and hence T ~ T O . For liquids, therefore, 



(68) p_ 

or, upon applying the Clausius-Mossotti law, 

(69) p-po = 



T>/T7 

In gases (67) is replaced by the relation p = T ~^, where R is the 

gas constant, T the absolute temperature, and M the molecular weight. 1 

2.26. Force on a Body Immersed in a Fluid. The results of our 

analysis may be summed up profitably by a consideration of the following 



Fluid 



FIG. 24. A solid body immersed in a fluid dielectric. 




problem. In Fig. 24, a dielectric or conducting body is shown immersed 
in a fluid. An electrostatic field is applied and we desire an expression 
for the resultant force on the entire body. Let us draw a surface Si enclosing 
the body and located in the fluid just outside the boundary S. On every 
volume element of the solid there is a force of density 

(70) f = pB-i^Ve + f", 

where f" is given by Eq. (40), and on each element of the surface of dis- 
continuity there is a traction given by (53) . Rather than evaluate the 
integrals of f and t over the volume and bounding surface S, we need only 

1 ABRAHAM, BECKER, and DOTTGALL, "The Classical Theory of Electricity and 
Magnetism," p. 98, Blackie, 1932. 



152 STRESS AND ENERGY [CHAP. II 

calculate the integral of t from Eq. (51) over the surface Si. Since Si 
lies in the fluid, ai = a 2 = T ~ and (51) reduces to 

(71) t - eE(E n) - I tf 2 n + ~ ~ #'n, 

where n is the unit normal directed from the solid into the fluid and e 
and r apply to the fluid. Now the integral of this traction over Si gives 
the force exerted directly by the field on the volume and surface of the 
body. But the field, as we have just seen, also generates a pressure 
(68) in the fluid which at every point on Si acts normally inward, i.e., 
toward the solid. The resultant force per unit area transmitted across 
Si is, therefore, equal to (71) diminished by (68), and the net force 
exerted by the field on the solid the force which must be compensated 
by exterior supports is 



= f I 



(72) F = II eE(E - n) - | # 2 n da. 

Thus to obtain the resultant force on an entire body we need not know 
the constants aj. and a% within either the solid or the fluid, for the forces 
with which they are associated are compensated locally by elastic stresses. 
Furthermore, since the fluid is in equilibrium, it is not essential that Si 
lie in the immediate neighborhood of the body surface S. The net force 
on the liquid contained between Si and an arbitrary enclosing surface S% 
is zero and consequently 

(73) f \*E(E n) - ~ E 2 nl da = f f cE(E n) - - E 2 nl da, 
Jsi L ^ J Jst L * J 

if we adhere to the convention that n is directed outward from the enclosed 
region. The surface S 2 rnay, therefore, be chosen in any manner that 
will facilitate integration, provided only that no foreign bodies are 
intersected or enclosed. 

In case the solid is a conductor, the field E is normal at the boundary. 
The resultant force on an isolated conductor is then 

(74) F = ^ f eE 2 nda = ~ f 6>Eda, 

^ JSi & JSi 

where co is the charge density on S. This last is true, of course, only for a 
surface Si just outside the conductor. 

It should be evident now that the force on one solid embedded in 
another can be calculated only when the nature of the contact over their 
common surface is specified; the problem i3 complicated by the fact that 



SEC. 2.27] NONFERROMAGNETIC MATERIALS 153 

tangential shearing stresses as well as normal pressures must be com- 
pensated by elastic stresses across the boundary. 

FORCES IN THE MAGNETOSTATIC FIELD 

2.27. Nonferromagnetic Materials. The analysis of Sec. 2.22 may 
be applied directly to the magnetostatic field in all cases where the rela- 
tion between B and H is of the form 

(1) B, = V MB* G* = 1, 2, 3), 

Wi 

with the components ^k of the permeability tensor functions possibly of 
position but independent of field intensity. The change in magnetic 
energy of an isotropic body resulting from a variation in M is given by 
Eq. (52), Sec. 2.17. 

(2) &T = 

and the generalization of this expression to an anisotropic body occupying 
the volume Vi is 



(3) 



= g f 



This variation may be expressed in terms of the components of strain, 
assuming a linear relation between the components of the permeability 
and strain tensors. 

3 3 

2 ^ 5e '-" ^ 



If we assume further that in the unstrained ztate the medium is isotropic, 
the coefficients 6& reduce to two, 



/e\ I, 

(5) Zh 

so that the variation in magnetic energy due to a pure strain is 

(6) ar = i f KbiffJ + biflj + biHJ) eu + (biH| + 



Se 12 



Equation (6) differs formally from (31), page 143, only in algebraic 
sign. Now the electrostatic energy U represents the work done in 



154 STRESS AND ENERGY [CHAP. II 

building up the field against the mutual forces between elements of 
charge. The magnetic energy T, on the other hand, represents work 
done against the mutual forces exerted by elements of current, plus 
the work done against induced electromotive forces. The work required 
to bring about a small displacement or deformation of a body in a mag- 
netic field is done partly against the mechanical forces acting on the body, 
partly on the current sources to maintain them constant. 1 When the 
work done against the induced electromotive forces is subtracted from 
the total magnetic energy, there remains the potential energy of the 
mutual mechanical forces, and this we found in Sec. 2.14 to be equal and 
opposite to T. 

(7) dU = -ST. 

The forces exerted by the field on the body within V\ are to be calculated 
from (6) with sign reversed. 

In complete analogy with the electrostatic case we now find that 
the body force exerted by a magnetic field on a medium which in the 
unstrained state is isotropic, whose permeability is independent of field 
intensity, which is free of residual magnetism, but which may carry a 
current of density J, is 

1 ^3 

_iJL 

2dXj 

If the medium is homogeneous and carries no current, (8) reduces to 
(9) f = -K&i + 



In a gas or liquid 61 = & 2 = r -~, and (8) can be written in the form 

OT 



(10) f = /.JXH- 

As in the electrostatic case, the body force whose components are 
given by (8) can be expressed as the divergence of a tensor 2 S, 

(11) f = V - 2 S, 
whose components are 




1 This remark does not apply if the source is a permanent magnet; in that case the 
energy of the field is not given by T - J/fi H dv. 



SBC. 2.28] FERROMAGNETIC MATERIALS 155 

(13) 



The force exerted by the magnetic field on the matter within a closed 
surface Si is, therefore, the same as would result from the application over 
Si of a force of surface density 

(13) t = 



Likewise the force per unit area exerted on a surface of discontinuity 
can be obtained by calculating (13) on either side of the surface. The 
result is the magnetic equivalent of (53) . A case of particular importance 
is that of a magnetic body immersed in a fluid whose permeability to a 
sufficient approximation is independent of density and equal to M O , the 
value in free space. Let the body be represented by the region (1) in 
Fig. 23 and the fluid by the region (2). The force per unit area on the 
bounding surface is then found to be 

(14) t = CLZ 



where the constants 61, 6 2 apply to the magnetic body and HI is measured 
just inside its surface. 

Not very much is known about the parameters 61 and 6 2 , although 
there are indications that they may be very large. They must, of course, 
be determined if the elastic deformation of a body is to be calculated. 
In most practical problems, fortunately, one is interested only in the net 
force acting on the body as a whole. The forces arising from deforma- 
tions are compensated locally by induced elastic stresses and conse- 
quently terms involving 61 and 6 2 will drop out. As in the electric case, 
the resultant force exerted by a magnetostatic field on a nonferromagnetic 
body immersed in any fluid is obtained by evaluating the integral 



(15) F = f f 

JSiL 



M H(H - n) - I tf 2 n da 

over any surface enclosing the body. 

2.28. Ferromagnetic Materials. The preceding formulas apply to 
ferromagnetic substances in sufficiently weak fields. If, however, the 
permeability \n depends markedly on the intensity of the field, the energy 
of a magnetic body can no longer be represented in the form of Eq. (2) 
or (3) and we must use in their place the integral derived in Sec. 2.17. 

(16) T = \ \ (Hi B-H-B l -H-B + 2 f H- dB ) dv. 

* JVi \ JO / 



156 STRESS AND ENERGY [CHAP. II 

An analysis of the volume and surface forces for such a case has been 
made by Pockels, 1 who also treats briefly the problem of magnetostriction. 
The phenomena of magnetostriction are governed, however, by several 
important factors other than the simple elastic deformation considered 
here. A specimen of iron, for example, which in the large appears 
isotropic, exhibits under the microscope a fine-grained structure. The 
properties of the individual grains or microcrystals are strongly aniso- 
tropic. Of the same order of magnitude as these grains but not necessarily 
identified with them are also groups or domains of atoms, each domain 
acting as a permanent magnet. In the unmagnetized state the orienta- 
tion of the magnetic domains is random and the net magnetic moment 
is zero. A weak applied field disturbs these little magnets only slightly 
from their initial positions of equilibrium. A small resultant moment is 
induced and under these circumstances the behavior of the iron may be 
wholly analogous to a polarizable dielectric in an electric field. As the 
intensity of the field is increased, however, the domains begin to flip over 
suddenly to new positions of equilibrium in line with the applied field, 
with a consequent change in the elastic properties of the specimen. A 
dilatation in weak fields may be followed by a contraction as the field 
becomes more intense, quite contrary to what would be predicted by the 
magnetostriction theory when applied to strictly isotropic solids. We 
are confronted here with a problem in which the macroscopic behavior of 
matter cannot be treated apart from its microscopic structure. 

FORCES IN THE ELECTROMAGNETIC FIELD 
2.29. Force on a Body Immersed in a Fluid. The expressions 

(1) su = -1 f E* fc cto, dT = I f ff 2 5 M dv, 

* J * J 

for the energy of a body in a stationary electric or magnetic field have 
been based in Sees. 2.10 and 2.17 on the irrotational character of the 
vectors E and H HI as well as upon their proper behavior at infinity. 
In a variable field these conditions are not satisfied and therefore (1) 
cannot be applied to the determination of the force on a body or an ele- 
ment of a body without a thorough revision of the proof. At best the 
analysis will contain some element of hypothesis, for our assumption that 

the quantities E and H represent the energy densities in an 

electromagnetic field is, after all, only a plausible interpretation of 
Poynting's theorem. 

1 POCKBLS, loc. cit., p. 369. 



+ J? 2 J n da. 



SEC. 2.29] FORCE ON A BODY IMMERSED IN A FLUID 157 

In Sec. 2.5 it was shown that the total force transmitted by an elec- 
tromagnetic field across any closed surface 2 in free space is expressed 
by the integral 

(2) F = f L(E n)E + 1 (B n)B - J ( , 

t/s L j^o * \ 

Subsequently we were able to demonstrate for stationary fields that if 
the surface 2 is drawn entirely within a fluid which supports no shearing 
stress, the net force is given by (2) if only we replace and MO by their 
appropriate values in the fluid. 

(3) F = e(E n)E + /i(H n)H - ~ 0=.E 2 + uH*}n da. 

J^ L Z \ 

This is the resultant force on the charge, current, and matter mthin 2. 
The matter within 2 need not be fluid and there may be sharp surfaces of 
discontinuity in its physical properties; the relations between E and D, 
and B and H, however, are assumed to be linear. 

Now since (2) is valid in the dynamic as well as in the stationary 
regimes, there is reason to suppose that (3) may be applied also to variable 
fields. It is not difficult to marshal support for such a hypothesis, 
particularly from the theory of relativity. However, the right-hand side 
of (3) must now be interpreted in the sense of Sec. 2.6 not as the force 
exerted by the field on the matter within 2 but as the inward flow of 
momentum per unit time through 2, We denote the total mechanical 
momentum of the matter within 2, including the ponderable charges, by 
G mcoh , and the electromagnetic momentum of the field by G f . Then 

(4) TJ (G mwh + G e ) = (E n)E + /t(H n)H - i (efi 2 + u# 2 )n da. 

/ 2 |_ 2t I 

The surface integral (4) may be transformed into a volume integral. 
For in the first place: 

l f _ i r 

2js 2J|; 

Then 

(6) %V(eE 2 ) = # 2 V + (D T)E + D X V X E. 

(7) (D . V)E = (DJK) + ^ t (DJS) + % 



(8) L fe (DxEx} + iy (DyE ^ + I (D * Ex) dv = (D ' n)S * da > 
and this obviously cancels the z-component of the vector *(E -n)E in 



158 STRESS AND ENERGY ICnAp. II 

(3). Proceeding similarly with the magnetic terms and then making 

use of the relations V X E = 9B/W, VXH = + J,V-D=p, we 

ot 

are led to 

(9) ^ ( G "<* + G.) = f [pE + J X B - | 
at jv L ^ 



The increase in the mechanical momentum is the result of the forces 
exerted by the field on charges and neutral matter. 



F, therefore, the right-hand side of (9) can be split into two parts identi- 
fied with G mec h and G e , the force can be determined. Just how this 
resolution is to be made is by no means obvious and various hypotheses 
have been suggested. 1 According to Poynting's theorem the flow of 
energy, even within ponderable matter, is determined by the vector 

(11) S = E X H joules/sec.-meter 2 . 

Abraham and von Laue take for the density of electromagnetic 
momentum 

1 

(12) g c = /xo^oS = -28 kg./sec.-meter 2 . 

c 

Then according to this hypothesis, the resultant force on the charges, 
currents, and polarized matter within S is 

(13) F = f ( P E + J X B - i 2 V _ 1 H 2 V/A + K * - 1 ^ ) dVj 

Jv \ & & c ot / 

or 

(14) F = f I (B n)E + /*(H - n)H - | (eB 2 + M ff 2 )n 1 da 

E X H dv. 



_llf 

c 2 d<J F ' 



Practically, the exact form of the electromagnetic momentum term 
is of no great importance, for the factor 1/c 2 makes it far too small to be 
easily detected. 

1 PAULI, Encyklopadie der mathematischen Wissenschaften, Vol. V, Part 2, 
pp. 662-667, 1906. 



SEC. 2.29] FORCE ON A BODY IMMERSED IN A FLUID 159 

On the grounds of (13) it is sometimes stated that the force exerted 
by an electromagnetic field on a unit volume of isotropic matter is 

(15) f = P E + JxB- * ' KmK '~ 



Such a conclusion is manifestly incorrect, for (15) does not include the 
forces associated with the deformation. As previously noted, these 
strictive forces are compensated locally by elastic stresses and do not 
enter into the integrals (13) and (14) for the resultant force necessary 
to maintain the body as a whole in equilibrium. 



CHAPTER III 
THE ELECTROSTATIC FIELD 

From fundamental equations and general theorems we turn our 
attention in the following chapters to the structure and properties of 
specific fields. The simplest of these are the fields associated with 
stationary distributions of charge. Of all branches of our subject, how- 
ever, the properties of electrostatic fields have received by far the most 
adequate and abundant treatment. In the present chapter we shall 
touch only upon the more outstanding of these properties and of the 
methods which have been developed for their analysis. 

GENERAL PROPERTIES OF AN ELECTROSTATIC FIELD 
3.1. Equations of Field and Potential. The equations satisfied by 
the field of a stationary charge distribution follow directly from Max- 
welPs equations when all derivatives with respect to time are placed 
Equal to zero. We have, then, at all regular points of an electrostatic 
field: 

(I) V X E = 0, (II) V - D - p. 

According to (I) the line integral of the field intensity E around any 
closed path is zero and the field is conservative. 

The conservative nature of the field is a necessary and sufficient con- 
dition for the existence of a scalar potential whose gradient is E. 

(1) E = -V<. 

The algebraic sign is arbitrary but has been chosen negative to conform 
with the convention which directs the vector E outward from a positive 
charge. Equation (1) docs not define the -potential uniquely, for there 
might be added to <t> any constant < without invalidating the condition 

(2) V X V(<#> + <o) ss 0. 

In Chap. II it was shown that the scalar potential of an electrostatic 
field might be interpreted as the work required to bring a unit positive 
charge from infinity to a point (x, y y z) within the field: 

(3) *(*,,*) = 

We shall show below that the field of a system of charges confined to a 
finite region of space vanishes at infinity. The condition that < shall 
vanish at infinity, therefore, fixes the otherwise arbitrary constant <<> 

160 



SEC. 3.1J EQUATIONS OF FIELD AND POTENTIAL 161 

Those surfaces on which < is constant are called equipotential sur- 
faces, or simply equipotentials. At every point on an equipotential the 
field intensity E is normal to the surface. For let 

(4) <t>(x, y, z) = constant 
be an equipotential, and take the first differential. 

(5) *-*' + S?'fr + *-0. 

The differentials dx, dy, dz, are the components of a vector displace- 
ment dr along which we wish to determine the change in <, and since 
d<f> = 0, this vector must lie in the surface, <t> = constant. The partials 
d<t>/dx, d<t>/dy, d<t>/dz, on the other hand, are rates of change along the 
x- y y-, z-axes respectively and as such have been shown to be the com- 
ponents of another vector, namely the gradient 

< 6 > ^-'fHfJ + kfr--*- 

d(f> is manifestly the scalar product of these two and, since this product 
vanishes, the vectors must be orthogonal. An exception occurs at those 
points in which the three partial derivatives vanish simultaneously. 
The field intensity is zero and the points are said to be points of 
equilibrium. 

The orthogonal trajectories of the equipotential surfaces constitute a 
family of lines which at every point of the field are tangent to the vector 
E. They are the lines of force. It is frequently convenient to represent 
graphically the field of a given system of charges by sketching the projec- 
tion of these lines on some plane through the field. Let ds represent a 
small displacement along a line of force, where 

(7) ds = i dx f + j dy f + k dz', 

the primes being introduced to avoid confusion with a variable point 
(z, y, z) on an equipotential. Then, since by definition the lines of 
force are everywhere tangent to the field-intensity vector, the rectangular 
components of ds and E(x', t/', z') must be proportional. 

(8) E x = \ dx', E v = X dy' y E, = X dz'. 
The differential equations of the lines of force are, therefore, 

/ON dx' dy' dz' 

1 ; E x (x', y', z'} E v (x', y', z'} E.(x', y', z') 

The relations between the components of D and those of E are almost 
invariably linear. If the medium is also isotropic, one may put 

(10) D = eE = -cV*, 



162 THE ELECTROSTATIC FIELD [CHAP. Ill 

whence, by (II), < must satisfy 

(11) V (eV<) = eV 2 < + Ve Vtf> = -p. 

In case the medium is homogeneous, < must be a solution of Poisson's 
equation, 

(12) V 2 = ~p. 

At points ot the field which are free of charge (12) reduces to Laplace's 
equation, 

(13) V 2 * = 0. 

The fundamental problem of electrostatics is to determine a scalar 
function <t>(x, y, z) that satisfies at every point in space the Poisson equa- 
tion, and on prescribed surfaces fulfills the necessary boundary condi- 
tions. A much simpler inverse problem is occasionally encountered: 
Given the potential as an empirical function of the coordinates repre- 
senting experimental data, to find a system of charges that would produce 
such a potential. The density of the necessary continuous distribution 
is immediately determined by carrying out the differentiation indicated 
by Eq. (12). There will in general, however, be supplementary point 
charges, whose presence and nature are not disclosed by Poisson's 
equation. At such points the potential becomes infinite and, inversely, 
one may expect to find point charges or systems of point charges located 
at the singularities of the potential function. The nature of these 
systems, or multipoles as they are called, is the subject of a later section, 
but a simple example may serve to illustrate the situation. 

Let the assumed potential be 



where r is the radial distance from the origin to the point of observation 
and a is a constant. By virtue of the spherical symmetry of this function, 
Poissons's equation, when written in spherical coordinates, reduces to 



on differentiation one finds for the density of the required continuous 
charge distribution 



The charge contained within a sphere of radius r is obtained by integrat- 
ing p over the volume, or 

(17) P dv = e~ r (ar + 1) - 1. 



SBC. 3.2] BOUNDARY CONDITIONS 163 

If the radius is made infinite, the total charge of the continuous dis- 
tribution is seen to be 

(18) 

But this is not the total charge required to establish the potential (14). 
For at r = the potential becomes infinite and we must look for a point 
charge located at the singularity. To verify this we need only apply 
(II) in its integral form. If q is a point charge at the origin, 



(19) 

the surface integral extending over a sphere S bounding the volume V. 

<*> *--- c (? 

(21) - J* D r da = (ar + 
and, hence, 

(22) q = +1. 

The potential defined by (14) arises, therefore, from a positive unit 
charge located at the origin and an equal negative charge distributed 
about it with a density p, the system as a whole being neutral. 

3.2. Boundary Conditions. The transition of the field vectors across 
a surface of discontinuity in the medium was investigated in Sec. 1.13 
and the results of that section apply directly to the electrostatic case. 
The two media may be supposed to meet at a surface S and the unit 
normal n is drawn from medium (1) into medium (2), so that (1) lies 
on the negative side of S, medium (2) on the positive side. Then 

(23) n X (E 2 - Ei) =0, n - (D 2 - DI) - , 

where <o is the density of any surface charge distributed over S. 

It will be convenient to introduce the unit vector t tangent to the 
surface S. The derivatives d</dn and d<t>/dt represent respectively the 
rates of change of $ in the normal and in a tangential direction. Then 
the boundary conditions (23) can be expressed in terms of the potential by 



From the conservative nature of the field it follows also that the potential 
itself must be continuous across S, for the work required to carry a small 
charge from infinity to either of two adjacent points located on opposite 
sides of S must be the same Hence 

(25) 0! = < 2 . 



164 THE ELECTROSTATIC FIELD [CHAP. Ill 

The two conditions 

are independent. 

Conductors play an especially prominent role in electrostatics. For 
the purposes of a purely macroscopic theory it is sufficient to consider 
a conductor as a closed domain within which charge moves freely. If 
the conductor is a metal or electrolyte, the flow of charge is directly 
proportional to the intensity E of the electric field: J = aE. Charge is 
free to move on the surface of a conductor but can leave it only under 
the influence of very intense external fields or at high temperatures 
(thermal emission). If the conductor is in electrostatic equilibrium all 
flow of charge has ceased, whence it is evident that at every interior 
point of a conductor in an electrostatic field the resultant field intensity E 
is zero, and at every point on its surface the tangential component of E is zero. 
Furthermore the electrostatic potential $ within a conductor is constant and 
the surface of every conductor is an equipotential. Let us suppose that an 
uncharged conductor is introduced into a fixed external field E . In the 
.first instant there occurs a transient current. According to Sec. 1.7, 
no charge can accumulate at an interior point, but a redistribution will 
occur over the surface such that the surface density at any point is w, 
subject to the condition 

(27) 

This surface distribution gives rise to an induced or secondary field of 
intensity EI. Equilibrium is attained when the distribution is such that 
at every interior point 

(28) Eo + E! = 0. 

Likewise, if a charge q is placed on an isolated conductor, the charge 
will distribute itself over the surface with a density o> subject to (28) 
and the condition 



(29) J co da = q. 

We shall denote the interior of a conductor in electrostatic equilibrium 
by the index (1) and the exterior dielectric by (2). Then at the surface S 

(30) EI = Di = 0, n X E a = 0, n D 2 = , 
or in terms of the potential, 

(31) <t> = constant, 2 ~^= -co. 



SEC. 3.3] GREEN'S THEOREM 165 

If a solution of Laplace's equation can be found which is constant over 
the given conductors, the surface density of charge may be determined 
by calculating the normal derivative of the potential. 

CALCULATION OF THE FIELD FROM THE CHARGE DISTRIBUTION 

3.3. Green's Theorem. Let V be a closed region of space bounded 
by a regular surface S, and let < and \l/ be two scalar functions of position 
which together with their first and second derivatives 1 are continuous 
throughout V and on the surface 8. Then the divergence theorem 
applied to the vector \t/V4> gives 

(1) f v V (W) dv = f s (tV<t>) - n da. 
Upon expanding the divergence to 

(2) V OAV0) = V\l/ V<t> + \f/V V0 = V$ V$ + ^V 2 $, 
and noting that 

(3) 



where d^/dn is the derivative in the direction of the positive normal, 
we obtain what is known as Green's first identity: 

(4) I V*-V<t>dv+ f *W<to= f*^dd. 

Jv Jv Js on 

If in particular we place ^ = <t> and let <f> be a solution of Laplace's 
equation, Eq. (4) reduces to 



f (V4)*dv= ( <t>^da. 
Jv Js on 



(5) 

Next let us interchange the roles of the functions <t> and ^; i.e., apply 
the divergence theorem to the vector <t>V\[/. 



(6) 



I V<j> V^ dv + I 0v V dv = f <t> ^ da. 
Jv Jv Js dn 



Upon subtracting (6) from (4) a relation between a volume integral and a 
surface integral is obtained of the form 

(7) 

known as Green's second identity or also frequently as Green's theorem. 

1 This condition is more stringent than is necessary. The second derivative of one 
function ^ need not be continuous. 



166 THE ELECTROSTATIC FIELD [CHAP. Ill 

3.4. Integration of Poisson's Equation. By means of Green's 
theorem the potential at a fixed point (z', y', z'} within the volume V can 
be expressed in terms of a volume integral plus a surface integral over S. 
Let us suppose that charge is distributed with a volume density p(x, y, z). 
We shall assume that p(x, y, z) is bounded but is an otherwise arbitrary 
function of position. An arbitrary, regular surface S is now drawn 
enclosing a volume 7, Fig. 25. It is not necessary that S enclose all 
the charge, or even any of it. Let O be an arbitrary origin and x = x', 
y = y^ z = z', a fixed point of observation within V. The potential 
at this point due to the entire charge distribution is <j>(x', y f , z'). For 




FIG. 25. Application of Green's theorem to a region V bounded externally by the surface 5 
and internally by the sphere Si. 

the function ^ we shall choose a spherically symmetrical solution of 
Laplace's equation, 

(8) t(x, y, z] x', y', z'} = -, 

where r is the distance from a variable point (x, y, z) within V to the 
fixed point (x f , y f , z'). 



(9) r = vV ~ *) 2 + (y f - yY + (*' - *) 2 - 

This function ^, however, fails to satisfy the necessary conditions of 
continuity at r = 0. To exclude the singularity, a small sphere of 
radius ri is circumscribed about (x', y', z 1 } as a center. The volume V is 
then bounded externally by S and internally by the sphere Si. Within 
V both <t> and ^ now satisfy the requirements of Green's theorem and 
furthermore V 2 \[/ = 0. Thus (7) reduces to 



the surface integral to be extended over both S and Si. Over the sphere 
Si the positive normal is directed radially toward the center (x'j y', z') 9 



Sac. 3.5] BEHAVIOR AT INFINITY 167 

since this ia out of the volume V. Over Si, therefore, 

(11) *$ - -**, 

UJ " 



Since ri is constant, the contribution of the sphere to the right-hand side 
of (10) ia 

*, 1 



If and d</>/dr denote mean values of <j> and 3<t>/dr on Si, this contribution 
is 



which in the limit ri > reduces to 4ir</>(z', y', 2')- Upon introducing 
this value into (10) the potential at any interior point (x', t/', z') is 



' 4ir Jv T 4ir Js [_r dn dn\rj 

or in terms of the charge density when the medium is homogeneous, 



i~t o\ j.// f f\ * I P i i * I I 1 v<p o il\l 

^loj <i>(x , y , z ) = - I ay + 7~ I I ~ <b I II da. 
4-Trc Jy T 4r js |_r dn dn \T/ ] 

In case the region V bounded by S contains no charge, (13) reduces to 
(14) *(*', y', *') = 1 



It is apparent that the surface integrals in (13) and (14) represent the 
contribution to the potential at (x r , y', z') of all charges which are exterior 
to S. If the values of < and its normal derivative over S are known, the 
potential at any interior point can be determined by integration. Equa- 
tion (14) may be interpreted therefore as a solution of Laplace's equation 
within V satisfying specified conditions over the boundary. The integral 

(15) ^'^- 

is a particular solution of Poisson's equation valid at (x', y', z 1 }] the 
general solution is obtained by adding the integral (14) of the homo- 
geneous equation V Z 4> = 0. If there are no charges exterior to S, the 
surface integral must vanish. 

3.6. Behavior at Infinity. Let us suppose that every element of a 
charge distribution is located within a finite distance of some arbitrary 



168 THE ELECTROSTATIC FIELD [CHAP. Ill 

origin 0, We imagine this distribution to be confined within the interior 
of a surface Si which also contains the origin 0. The distances r, ri and 
jB are indicated in Fig. 26. 

(16) r = (# 2 + r\ - 2riR cos 0)*. 

The potential at any point P(x' J y', z') outside Si is then 

(17) 6(x' v' z')- p(x > y > *> dv 
(17) t(x , y , z ) - 



As P recedes to infinity, the terms r\ and 2riR cos 6 become negligible 




FIG. 26. Figure to accompany Sec. 3.5. 

with respect to R 2 , and in the limit as R > oo we find 



(18) 



f 

J 7i 



where q is the total charge of the system. A potential function is said 
to be regular at infinity if R<fr is bounded as R > <*> . The field intensity 
E = v< at great distances is directed radially from and the function 
R*\E\ is bounded. 

All real charge systems are contained within domains of finite extent 
and their fields are, therefore, regular at infinity. Frequently, however, 
the analysis of a problem is simplified by assuming the external field to 
be parallel. Such a field does not vanish at infinity and can only arise 
from sources located at an infinite distance from the origin. 

It is important to note that a closed surface S divides all space into 
two volumes, an interior Vi and an exterior V 2 , and that if the functions 
<t> and \// are regular at infinity Green's theorem applies to the external 
region F 2 as well as to Vi. For certainly the theorem applies to a closed 
region bounded internally by S and externally by another surface S 2 . 

If now S 2 recedes towards infinity, the quantities \f/ and <t> ~ vanish 

on an 



SEC. 3.6] COULOMB FIELD 169 

as 1/r 3 and the integral over 82 approaches zero. Consequently 

(19) (W ~ *W * - , - , da, 



but the positive normal at points on S is now directed out of V z and 
therefore into V\. Let us suppose, for example, that a charge system is 
confined entirely to the region V i within S. The potential at any point 
(z', y', z') in 7 2 outside S may be calculated from (17), or equally well 
from a knowledge of <t> and d^/dn on S, applying (14) as indicated in 
Fig. 27. 




FIQ. 27. Application of Green's theorem to the exterior Vt of a closed surface 5. 

3.6. Coulomb Field. According to (18) the potential of a charge q in 
a homogeneous medium at distances very great relative to the dimensions 
of the charge itself approaches the value 

(20) ^ ; ^ 20 

If q is located at (x, y, z), the distance from q to the point of observation 
(*', y', z') is 



(21) r = V<X - *) 2 + (y' - yY + (*' - z) 2 . 

Let now r represent a unit vector directed from the source; i.e., from 
(x, y, z) towards (x', y', z'). Then 

(22) ^ 

where the prime above the gradient operator denotes differentiation 
with respect to the variables (x', y', z') at the point of observation. 
The field intensity at this point is 

(23) E( 

The field of a point charge is inversely proportional to the square of the dis- 
tance and is directed radially outward when q is positive. This is the law 



170 THE ELECTROSTATIC FIELD [CHAP. Ill 

established experimentally by Coulomb and Cavendish which is usually 
taken as a point of departure for the theory of electrostatics. The term 
"point charge " is employed here in the sense of a charge whose dimen- 
sions are negligible with respect to r. Mathematically one may imagine 
the dimensions of q to grow vanishingly small while the density p is 
increased such that q is maintained constant. In this fashion a point 
singularity is generated and (23) is then valid at all points except r = 0. 
There is no reason to believe that such singularities exist in nature, but 
it is convenient to interpret a field at sufficient distances as that which 
might be generated by systems of mathematical point charges. We 
shall have occasion to develop this concept in the subsequent section. 

The potential at (x f , y', z') is obtained by integrating the contribu- 
tions of charge elements dq = p dv over all space. 

(24) 



(25) dE = - -L p(x, y, z)V 0) dv = ~ P (x, y, z) V (^ dv. 

The field intensity due to a complete charge distribution in a homo- 
geneous, isotropic medium is, therefore, 



(26) 



- 4^ J 



3.7. Convergence of Integrals. The proof of convergence of the 
integrals for potential and field intensity is implicit in the method by 
which they have been derived, but an alternative treatment will be 
described which may serve as a model for proofs of this kind. Since the 
element of integration at (a;, y, z) can coincide with (x', y', z'), with the 
result that the integrand becomes infinite at this point, it is not obvious 
that the integral has a meaning. It will be shown that, although an 
improper integral of this type cannot be defined in the ordinary manner 
as the limit of a sum, it can by suitable definition be made to converge 
absolutely to a finite value. 

Let the point Q(x', y', z') at which the field is to be determined be 
surrounded by a closed surface $ of arbitrary shape, thus dividing the 
total volume V occupied by charge into two parts: a portion V\ repre- 
senting the volume within S and a portion V% external to S. Throughout 
F 2 the integral of Eq. (26) is bounded and consequently the charge 
outside S contributes a finite amount to the resultant field intensity at Q, 
a contribution E 2 , 

(27) E 2 (*', y', *0 = ~ P (x, y, *) V ( dv. 



SBC. 3.7] CONVERGENCE OF INTEGRALS 171 

The contribution of the charge within S to the field intensity at Q may 
be called Ei. If now the field intensity E = EI + E 2 at Q is to have a 
definite significance, it is necessary that EI vanish in the limit as S 
shrinks about Q. A moment's reflection will indicate that this is indeed 
the case for, although the denominator of the integrand vanishes as the 
square of the distance, the element of charge in the numerator is propor- 
tional to the volume Vi which vanishes as the cube of a linear dimension. 
It has been tacitly assumed that the charge density p is finite at every 
point in the volume V. There exists, therefore, a number m such that 
\p\ < m, and \p/r\ < m/r, at every point in V. It follows, furthermore, 
for this upper bound m that 



(28) P V(5 

and 

dv 



LOG) 



The surface S bounding the volume Vi is of arbitrary form, and so, to 
avoid the awkwardness of evaluating the integral over this region, a 
sphere of radius a concentric with Q is circumscribed about Vi. The 
volume element of integration is positive; consequently the integral 
extended throughout Vi must be less than, or at the most equal to, the 
integral extended throughout the volume enclosed by the circumscribed 
sphere. 



which evidently vanishes with a. The contribution of the charge within 
S to the field at Q becomes vanishingly small as S shrinks about Q and 
hence (26) converges for interior as well as exterior points of a charge 
distribution. 

The potential integral 



(31) 



>'- "'>*'> = i^ J ; 



is also an improper integral when the point of observation is taken within 
the charge but, since the denominator of the integrand vanishes only 
as the first power of r, the proof above holds here a fortiori. If p is a 
bounded, integrable function of position, (31) is a continuous function 
of the coordinates z', y', z', has continuous first derivatives, and satisfies 
the condition E = V'$ everywhere. It can be shown, furthermore, 



172 



THE ELECTROSTATIC FIELD 



[CHAP. Ill 



that if p and all its derivatives of order less than n are continuous, the poten- 
tial <t> has continuous derivatives of all orders less than n + I. 1 



EXPANSION OF THE POTENTIAL IN SPHERICAL HARMONICS 

3.8. Axial Distributions of Charge. We shall suppose first that an 
element of charge q is located at the point z = f on the 2-axis of a rec- 
tangular coordinate system whose origin is at 0. We wish to express the 
potential of q at any other point P in terms of the coordinates of P with 
respect to the origin 0. The rectangular coordinates of P are x, y, z, 
but since the field is symmetric about the z-axis it will be sufficient to 

locate P in terms of the two polar coordi- 
nates r and 0, Fig. 28. The distance from 
q to P is r 2 and the potential at P is, therefore, 




(1) 



1 



the medium being assumed homogeneous 
and isotropic. 

(2) r 2 = (r 2 + f 2 - 2rf cos 0)*. 



FIQ. 28. Figure to accompany 
Sec. 3.8. 



There are now two cases to be considered. 
The first and perhaps less common is that 
in which P lies within a sphere drawn from as a center through f . 
Then r < f and we shall write 



(3) 



The bracket may be expanded by the binomial theorem if 



M-j. 



1. 



If furthermore 



2 cos B 



< 1, the resultant series converges abso- 



lutely and consequently the various powers may be multiplied out and 
the terms rearranged at will. If the terms of the series are now ordered 
in ascending powers of r/f , we find 



(4) f- - r 



1 See for example KELLOGG, " Foundations of Potential Theory," Chap. VI, 
Springer, 1929, or PHILLIPS, "Vector Analysis," pp. 122 #, Wiley, 1933. 



SEC. 3.8] AXIAL DISTRIBUTIONS OF CHARGE 173 

which shall be written in the abbreviated form 

(5) ~ 

The coefficients of r/f are polynomials in cos and are known as the 
Legendre polynomials. 

P (cos 0) = 1, 
PI(COS 0) == cos 0, 
^ P 2 (cos 0) = i(3 cos 2 - 1) = (3 cos 20 + 1), 

P 8 (cos 0) = i(5 cos 3 3 cos 0) = |(5 cos 30 + 3 cos 0). 

The absolute value of the coefficients P n is never greater than unity; 
hence the expansion converges absolutely provided r < |f|. 

In the second case P lies outside the sphere of radius f , such that 
r > f . The corresponding expansion is obtained by interchanging r 
and fin (3) and (5). 

(7) ^ = 

(8) ' 

This last result may be obtained in a slightly different manner. 
Consider the inverse distance from the point z = f to P as a function of f 
and expand in a Taylor series about the origin, f = 0. 

1 

(9) /(f) = = (r 2 + f 2 - 2rf cos 0)~*, 



Now in rectangular coordinates, r 2 is 
(11) r 2 = (x* + y* 

and 



Hence^ 



and, since /(O) = 1/r, we have 



174 THE ELECTROSTATIC FIELD [CHAP. Ill 

The potential at a point P outside the sphere through q can be written 
in either of the forms 



47T6 1 f*l ~ 47T6 

n-0 n-0 



from which it is apparent that 

P n (cos 6} _(-!)* 3* (l\ 

" ~^r a^ vv 



Finally, let us suppose that charge is distributed continuously along 
a length I of the z-axis with a density p = p(f). The potential at a 
sufficiently great distance from the origin is 

(17) <t>(r, 6) - * - ' p(f)f " * ^' ( r > <) 



The leading term of this expansion, 



where g is now the iotoZ charge on the line, is evidently the Coulomb 
potential of a point charge q located at the origin. However, the density 
may conceivably assume negative as well as positive values such that 
the net charge 



(19) 



= r 

Jo 



is zero. The dominant term approached by the potential when r ~2> I is 
then 

/ 9n s , _ 1 Pi(cos 6} C l , . , _ p cos e 

(20) fc - ^ p J P(f)f * - j^ r- 

The quantity 

(21) p = J o P (f)fdr 

is called the dipole moment of the distribution. In general, we shall 
write 



and define 

(23) P w = J o 

as an axtaZ multipole of nth order. 



SEC. 3.9] THE DIPOLE 175 

3.9. The Dipole. In order that the potential of a linear charge dis- 
tribution may be represented by a dipole it is necessary that the net 
charge be zero the system as a whole is neutral and that the distance 
to the point of observation be very great relative to the length of the line. 
We have seen that the potential < is that which would be generated by a 
mathematical point charge located at the origin. < has a singularity 
at r = 0, for a true point charge implies an infinite density. We now 
ask whether a configuration of point charges can be constructed which 
will give rise to the dipole potential fa. 

Let us place a point charge +# at a point z = I on the z-axis and an 
equal negative charge q at the origin. According to Eq. (8) the 
potential at a distant point is 



/0 ,x 
(24) 



. q (\ 1\ ql cos , . , , , 

4 = JL ^_ - _J = JL- _ _ + higher order terms. 



The product p = qlis evidently the dipole moment of the configuration 
Suppose now that I > 0, but at the same time q is increased in magnitude 
in such a manner that the product p remains constant. Then in the 
limit a double-point singularity is generated whose potential is 



everywhere but at the origin. A direction has been associated with a 
point. The dipole moment is in fact a vector p directed, in this case, 
along the z-axis. The unit vector directed along r from the dipole 
towards the point of observation is again r, and the potential is, 
therefore, 1 

1 o r t 

(26) <l>(x, T/, z) == ~ *-~ == j p ' 

The field of a dipole is cylindrically symmetrical about the axis; 
hence in any meridian plane the radial and transverse components of 
field intensity are 

d(j> 1 p cos 

(27} r 

E = _i** = 1 P sin 

9 r 36 4ire r 3 



1 In (26) r = \/a; 2 -f- 2/ 2 4- z* and V(l/r) implies differentiation at the point of 
observation. If the dipole were located at (x, y, z) and the potential measured at 

(r/, y' f 2'), we should have (x', y', z'} - p V' ( - ) - 4--^- P V (- ) See 

4rr \r/ 4re \f/ 



Eq. (22), p. 169. 



176 



THE ELECTROSTATIC FIELD 



[CHAP. UI 



The potential energy of a dipole in an external field is most easily 
determined from the potential energies of its two point charges. Let a 
charge +g be located at a point a, and a charge q at b displaced from 
the first by an amount 1, in an external field whose potential is (j>(x, y, z). 
The potential energy of the system is then 



(28) 



U = q<t>(a) - 



or as 6 > a, 

(29) U = q d<t> = ql V< = p E = pE cos 9, 

where 6 is the angle made by the dipole with the external field E. 




FIG. 29. Lines of force in a meridian plane passing through the axis of a dipole p. 



The force exerted on the dipole by the external field is equal to the 
negative gradient of U when the orientation is fixed. 



(30) 



F = V(p 



On the other hand a change in orientation at a fixed point of the field also 
leads to a variation in potential energy. The torque exerted on a dipole 
by an external field is, therefore, 



(31) 

or vectorially, 

(32) 



W 

__ = - pE 

T = p X E. 



3.10. Axial Multipoles. Let us refer again to Eq. (17) for the poten- 
tial of a linear distribution of charge. This expansion is valid at all 
points outside a bphere whose diameter is the charged line. Now the 
first term <<> of the series is just the potential that would be produced 
by a point charge q located at the center of the sphere. The second term 
<f>i represents the potential of a dipole p located at the same origin. We 
shall show that the remaining terms <t> n may likewise be interpreted 
as the potentials of higher order charge singularities clustered at the 
center. 



SBC. 3.10] AXIAL MULTIPOLES 177 

The dipole, whose moment we now denote as p (1) , was constructed 
by placing a negative charge q at the origin and a positive charge +q 
at a point z - Z along the axis. The two are then allowed to coalesce 
such that p (1) = ql Q remains constant. The potential of the resultant 
singularity is 

(33) ^ 

The singularity of next higher order is constructed by locating a dipole 
of negative moment p (1) at the origin and displacing from it another 
equal positive moment by the small amount li. For the present we 
confine ourselves to the special case wherein the axes of both dipoles 
as well as the displacement li are directed along the 2-axis, The potential 
of the resultant configuration is 

(34) <#>2 = 4>i + 
or by virtue of (12) 

(35) *._L 

An axial quadrupole moment is defined as the product 

(36) p( 



The mathematical quadrupole is generated by letting Z > 0, l\ > 0, 
q > oo such that the product (36) remains finite. The potential of this 
configuration is then strictly 

-$ 

at all points excluding r = 0. 

By induction one constructs charge singularities, or multipoles, of 
yet higher order. In each case a multipole of order n 1 and negative 
moment p< w - l > is located at the origin and an equal positive multipole 
p(n-i) displaced from it by l n . In the general case to be dealt with below, 
the displacements are arbitrary in direction; if, as in the present special 
case, all are along the same straight line, the multipole is said to be axial. 
The potential is given approximately by the first term of a Taylor series, 
which again by (12) may be written 



The n-pole moment is defined as the limit of the product 
(39) p<"> = np< 



178 



THE ELECTROSTATIC FIELD 



[CHAP. Ill 



when l n - and p<*-*> oo in such a manner that p (n ~ i n n remains 
finite. The potential of a point charge of nth order is then 



(40) 



(-1)- 

nl 



(n = 0, 1, 



The potential of an arbitrary distribution of charge along a line is identical, 
outside a sphere whose diameter coincides with the line, with the potential 
of a system of multipoles located at the origin. The moments of these 
multipoles can be determined from the charge distribution by (23). 

3.11. Arbitrary Distributions of Charge. We shall consider next a 
charge which is distributed in a completely arbitrary manner in any 
finite region of a homogeneous, isotropic dielectric, or upon the surfaces 
of conductors embedded in such a dielectric. The region containing 
charge lies within a sphere of finite radius R drawn about the origin O 
of a coordinate system. We wish to express the potential of the dis- 

tribution at a point P outside the sphere 

in terms of the coordinates of P with 

respect to 0. 

The rectangular coordinates of P 

are x, y, z and its polar coordinates 

r, 0, $, where 

(41) x = r sin 6 cos \//, 

y r sin sin ^, z r cos 6. 

We calculate first the potential at P of 
an element of charge dq = p dv located 

Figure to accompany Sec. 3.11. at the var i a bl e point , 77, f. If r,, 

Fig. 30, is the distance from dq to P, the contribution of this element is 



P <x, y, z) 




(42) d<t> 



JL 
47T6 



- 



) 2 + (y i?) 2 + (z f) 2 

The denominator of this function can be expanded about the origin in 
powers of , 77, f exactly as in Eq. (14), and one obtains 



(43) 






dy dz 



dz dx r 



The expansion converges provided ri == \/ 2 + ^7 2 + f 2 <H,r > R. 

Next we integrate (43) over the entire charge distribution; i.e., with 
respect to the coordinates , 77, f, noting that r = \/x 2 + y 2 + z* is 



SEC. 3.12J GENERAL THEORY OF MULTIPOLES 179 

independent of this integration. Once again we shall write <j> as the 

00 

sum of partial potentials <f> nj $ = < n . Then the first term of the 



expansion is 

1 <n(0) 

(44) '-iV' 
where 

(45) pt = q = f P (, ,, f) <fo 

is the total charge. At sufficiently large distances relative to R the potential 
of an arbitrary distribution of charge may be represented approximately 
by the Coulomb potential of a point charge located at the origin. 

The second term, which is dominant when the net charge is zero, is 



(46) _ ^--. 

where p (1) is a vector dipole moment whose rectangular components are 

(47) pp = f P dv, pp = fprj dv, p< = f pf dv. 

The potential of any distribution whose net charge is zero may be approxi- 
mated at large distances by the potential of a dipole located at the origin 
whose components are determined by (47). 

In like manner the partial potential < 2 arises from a quadrupole whose 
components are 

p> = f P ? dv, p$ = f py* dv, p% = f pf 2 dv, 
(48) 

P { $ = f Pti dv, p% = J piyr &, pii* = J pf cfo. 

Whereas the dipole moment is a vector, the quantities defined in (48) 
constitute the components of a tensor of second rank. The multipole 
moments of higher order determined from subsequent terms of the 
expansion (43) are likewise tensors of higher rank. 

3.12. General Theory of Multipoles. Let l t be a vector drawn from 
the origin to the point Q( , 77, f ) whose direction cosines with the coor- 



dinate axes #, y, z are respectively a t -, ft, 7*. Let ~-~ /($, ?/, 2) be the 

potential at the point P(x, y, z) due to a charge singularity p<*> of any 
order located at the origin. If now a charge singularity of equal magni- 
tude but positive sign be placed at the outer extremity of the vector li, 
the potential at P due to the pair of multiple points will be 

(49) ^ x = 2L!- [/, ( X - vfc y - p& Z - y J.) - /fa y Z )] 





180 THE ELECTROSTATIC FIELD [CHAP. Ill 

for it is to be noted that the change in potential at (#, y, z) arising from a 
displacement of the source from the origin to a point Q whose coordinates 
are = aZ, rj (Hi, f = y l l i , is identical with that resulting from an 
equal but opposite displacement of P to a point (x , y y, z f) 
during which the source remains fixed at the origin. Expanding by 
Taylor's theorem, we have 

(50) ^(x, y, z) = -?p (a, g + ft ^ + 7*f) + terms of higher 

order in /. 

The expression in parentheses is evidently the derivative of /, with 
respect to the direction specified by the vector l t . 



-. - *, 

__ a< __ 7 ,_. 

The multipole moment of order i + 1 is again defined in terms of the 
moment of order i by the limit of 

(52) p<*H> = (i + l) p >Zi, (i = 0, 1, - - ), 

as li > and pW > oo in such a manner that their product remains 
finite. The potential at any point (x, y, z) due to a multipole of moment 
located at the origin is then 



A single point charge may be considered as a singularity of zero 
order and " moment" p (0) = q. 

(54) *.-iS 

The potential of a dipole oriented in any direction specified by the vector 

lo is consequently 



The quadrupole is next generated by displacing the dipole parallel to 
its axis in a direction specified by the vector li. For its potential we find 
from (53) and (55) 



Thus by induction the potential of the general multipole of nth order is 

A _ (- !)?<> a" 

<t, n - __ -^ ^ Wi . . . 



SBC. 3.12] 



GENERAL THEORY OF MULTIPOLES 



181 



The construction of a quadrupole and an octupole is visualized in Figs 
31a, 316. 

The result of applying the operator 

XfQN C7 I O I ^ 

n times to the function - = (x 2 + y* + z 2 )-* must be of the form 
(59) < n = - p(*) _JL, 



where F n is a function of the cosines of the angles m between r and the 
n axes It, and of the direction cosines a iy fr, y t of the angles made by the 




FIG. 3 la. Quadrupole. 



FIG. 316. Octupole. 



vectors 1< with the coordinate axes. But the angles /* are related to 
the spherical coordinates 0, $ of P by the equation 

(60) cos m = (on cos \l/ + fa sin ) sin 6 + 7; cos 0, 

which may be demonstrated by calculating the distance QP = r 2 , Fig. 
30, both in terms of n = Z<, r, and the angle m, and in terms of its projec- 
tions on the coordinate axes. Thus Y n is a function of the angles 6 and ^, 
and hence of position on the surface of a unit sphere. For the dipole 
term we have 



702). 

Upon replacing rectangular by spherical coordinates, Eqs. (41), this 
becomes 



(62) 

or 

(63) 



4^ 72" 



_ 



cos 



cos \l/ + /3 sin \l/) sin ^ + 70 cos 0], 
$ + fa sin ^)P}(cos 0) + 70 PJ(cos 8)], 



182 THE ELECTROSTATIC FIELD [CHAP. Ill 

where the functions P(COS 0) are the associated Legendre functions 
defined by 

(64) P?(cos 0) = sin" e ^~^ P(cos 0). 

Between the four constants p (1) , a 0} /5 0; 70, there exists one relation, 
namely + Pi + To = 1, and there remain, consequently, three arbi- 
trary parameters which determine the magnitude and orientation of the 
dipole. 

Proceeding to the quadrupole term, we obtain 



+ g (jSori + ]8iYo) sin ^ + (TC<*I + 71^0) cos ^ U sin 20 

+ ]; (o/3i + i/5o) sin 2^ + j (a i + o0i) cos 2^ 3 sin 2 0\, 
or, with reference to (64), 



(66) <#> 2 = j -5- [a 2 oP(cos 0) + (a 2 i cos ^ + 6 2 i sin 

+ (a 22 cos 2^ + 62-2 sin 2^)Pi(cos 8)]. 

Between the six direction cosines there are two relations of the form 
af + ($\ + y\ = 1; consequently there are, together with the arbitrary 
moment p (2) , five completely arbitrary constants which are sufficient to 
determine the magnitude and orientation of the quadrupole. 

A solution of Laplace's equation is called a harmonic Junction. Since 
the coordinates r, 0, \J/ are arbitrary, each term < n in the expansion of j> 
must itself satisfy Laplace's equation and is, therefore, harmonic. Gen- 
eralizing, we conclude that the harmonic function </> n representing the 
potential of a multipole of nth order is a homogeneous polynomial of nth 
degree in , y, 2. Upon transformation to spherical coordinates r, 0, ^, 
the function < n can be expressed in the form (59), where the spherical 
surface harmonic F n (0, $} is now explicitly 

n n 

(67) Y n (0,f) 

The harmonic of nth order contains 2n + 1 arbitrary constants sufficient 
to determine the moment and orientation of the axes of the corresponding 
multipole. If charge be distributed in arbitrary manner within a sphere 
of finite radius, the potential at all points outside the sphere must be 
harmonic and it must furthermore be regular at infinity. These condi- 



Sue. 3.13] INTERPRETATION OF THE VECTORS P AND H 183 

tions are satisfied by 

(68) 



If the charge distribution is known, the components of the multipole 
moments are determined as in Sec. 3.11; if the distribution is unknown 
but an external field is specified, these constants are obtained from the 
boundary conditions as we shall shortly see. 

DIELECTRIC POLARIZATION 

3.13. Interpretation of the Vectors P and n. Although the two vec- 
tors E and D are sufficient to characterize completely an electrostatic 
field in any medium, it is convenient to introduce a third vector P 
defined in Sec. 1.6 as the difference of these two fields, 

(1) P = D - 6 E. 

The vector P vanishes in free space and therefore is definitely associated 
with the constitution of the dielectric. The divergence equation then 
becomes 

(2) V-E - -(p ~ V-P), 

CD 

from which one concludes that the effect of a dielectric on the field may 
be accounted for by an equivalent charge distribution whose volume 
density is 

(3) p'=-V-P. 

At every interior point of the dielectric the potential satisfies a modified 
Poisson equation, 

(4) V*+ = -!(p + p'). 

*o 

The validity of (4) is no longer contingent upon the isotropy and homo- 
geneity of the medium. If, however, there are surfaces of discontinuity, 
such as the boundaries of a dielectric, the vector D is subject to the 
condition 

(5) n (D 2 - DO = co, 

(6) n-(E 2 -Ei) = -co - -n-(P 2 - PJ = ~(co + co'). 

CO Co Q 

The primary sources of an electrostatic field are the "real" charges whose 
volume and surface densities are respectively p and co. The effect of 
rigid dielectric bodies on the potential may be completely accounted for 
by distributions of induced, or "bound," charges of volume density p' 



184 THE ELECTROSTATIC FIELD [CHAP. Ill 

and a density 

(7) a/ = -n (P 2 - Pi) 

over each surface of discontinuity, the unit vector n being drawn from 
medium (1) into (2). If (1) is dielectric and (2) free space, the induced 
surface charge is simply a/ = n PI, where n is the outward normal. 
In the case of a transition from a dielectric (2) to a conductor (1) bearing 
a free surface charge o>, there results a net surface charge o> n P 2> 
since PI vanishes within the conductor. 

The potential at any fixed point either within a dielectric or exterior 
to it can be expressed by 



The surface integrals are to be extended over all surfaces of discontinuity. 
It need scarcely be remarked that this analysis in most cases is purely 
formal, for in order to know P it is usually necessary to calculate first 
the potential. 

The physical significance of the vector P becomes apparent if we trans- 
form (8) with the help of the identity 

i_p =v 



The dielectric may be resolved into partial volumes Vt bounded by 
surfaces Si of discontinuity. For each partial volume 

(10) 

^ } 



f V.(*U = f 
JVi \rj Js 



On summing these integrals over the partial volumes and remembering 
that Ui is directed out of V* and hence in to a contiguous Vkj one finds 
that (8) reduces to 



(11) 

\ j 



(x f , y', *') = -r^- (~dv + ^ ff^da + T^- f 
,*, / 47rcoJ r 47ToJ r 47re J 



The third integral is now clearly the potential produced by a continuous 
distribution of dipole moment, and P is, therefore, to be interpreted as 
the dipole moment per unit volume, or polarization of the dielectric. 
According to (11) a dielectric body in the primary field of external 
sources gives rise to an induced secondary field such as is generated by a 
distribution of dipoles of moment P per unit volume. 

In a purely macroscopic theory this is really all that can be said about 
the polarization vector. Macroscopically a charged conductor behaves 



SBC. 3.14J VOLUME DISTRIBUTIONS OF CHARGE 185 

as though the charge were distributed with a continuous density, though 
in fact we know it to be constituted of discrete electronic charges. Like- 
wise, neutral matter under the influence of an external field acts as though 
dipoles were distributed as a continuous function of position. Now we 
know that neutral atoms and molecules are in fact constituted of equal 
numbers of positive and negative elementary charges. A microscopic 
electrodynamics must show how the electrical moments of each atom 
or at least their most probable values can be calculated. Then by a 
suitable process of averaging over large-scale volume elements the polari- 
zation per unit volume will be determined from the atomic moments 
and the transition effected from microscopic to macroscopic quantities. 

A word at this point is in order concerning the physical significance 
of the Hertz vector n introduced in a more general connection in Sec. 
1.11. We shall suppose that the density of free charge p within the 
dielectric is zero. If the potential <f> is expressed as the divergence of a 
vector function IJ, 

(12) = -V-n, 

then Poisson's equation assumes the form V ( V 2 n + - P ) = and 

\ / ' 

this condition will certainly be satisfied if 

(13) V 2 H = -i P. 

60 

The vector n is determined from the polarization by evaluating the 
integral 

(14) n (*', </', *') = JL- J p(x >v> ^ dv, 

whence n is sometimes referred to as the polarization potential. 

DISCONTINUITIES OF INTEGRALS OCCURRING IN POTENTIAL THEORY 

3.14. Volume Distributions of Charge and Dipole Moment. Accord- 
ing to the foregoing analysis, a conductor is simply a region of zero field 
bounded by a surface bearing a layer of charge of density w, while a rigid 
dielectric may be represented either by an equivalent volume distribution 
of density p' = V P bounded by a surface layer a/, or as a region 
occupied by a continuous distribution of dipoles whose moment per 
unit volume is P. The boundary conditions, which heretofore have 
been deduced from the field equations, must evidently follow directly 
from the analytic properties of the integrals expressing the potential 
and its derivatives in terms of volume and surface densities of charge 
and dipole moment. 



186 THE ELECTROSTATIC FIELD [CHAP. Ill 

In Sec. 3.7 it was established that the improper integrals 



(i) 

(2) 



E(*', y', ^') = 4 J P(X, y, z)V (i 



da, 



are convergent and continuous functions of x f , y', z r provided only that the 
charge density is bounded and piecewise continuous; i.e., that the volume 
occupied by the charge can be resolved into a finite number of partial 
volumes within each of which p is finite and continuous. Thus on either 
side of a surface bounding a discontinuous change in p, <t> and E will have 
the same values, although the derivatives of E will in general be dis- 




'Jfto. 32. Transition of potential and field intensity at the surface of a uniform spherical 

charge of unit radius. 

continuous. Consider, for example, a spherical distribution of constant 
density p and radius a. At a point inside, the field E is found most 
simply by taking account of the spherical symmetry and applying the 

Gauss law, I E n da = I p dv. At a distance r from the origin, 
r < a } we have 

(3) E = ^ p r = 2 ~V ( r < a )> 

where q is the total charge contained within the sphere. Likewise at any 
external point 

(4) E- 



(5) 



The potential is calculated from the integral <t> = P E dr, giving 

0= -JL- f r ! drs l 2, ( r >o), 

^ 47T J co r 2 47T6o T 

* - -i*- f* V - /- frdr - =*-(? - ^, (r < a). 

v 4ir Jr 2 4ir Ja Sireo \a a 3 / 



SBC. 3.15] SINGLE-LAYER CHARGE DISTRIBUTIONS 187 

In Fig. 32, < and its first derivative E are plotted as functions of r. 
Note that although E passes continuously through the surface r = a, 
its slope the second derivative of <j> suffers an abrupt change because 
of the failure in continuity of the density p. 

Consider next a volume distribution of dipole moment of density P. 
The potential at any exterior point is 

(6) ^', 2/ ' ) ,') 



This can be resolved into three integrals of the type 

1 f d (\\ 1 d CP 

(7) 01 "i^J Px te\r) dv== ~4^;?J ~^ dv ' 

From what has just been proved regarding (1) it follows that (6) converges 
within the dipole distribution as well as at points exterior to it, and is a 
continuous function of position provided only P(x, y, z) is bounded and 
piecewise continuous. Since a dielectric body is equivalent to a region of 
dipole moment, we have here proof of the continuity of the potential 
across a dielectric surface without recourse to an energy principle. 

Across a surface of discontinuity in P, the normal derivative of (6), 
and consequently the normal component of E, is discontinuous. The 
magnitude of this discontinuity can be determined most readily by 
reverting from the volume distribution of moment to the equivalent 
volume and surface-charge distribution defined in Sec. 3.13. The vector 
E passes continuously into the volume charge, but we shall now show that 
its normal component suffers an abrupt change in passing through a 
layer of surface charge. 

3,15. Single-layer Charge Distributions. Let charge be distributed 
over a surface >S with a density co which we shall assume to be a bounded, 
piecewise continuous function of position on S. The potential at any 
point not on S is 

(8) 

where r as usual is drawn from the charge element co(x, t/, z} da to the 
point of observation. If now (x', ?/', z 1 ) lies on the surface, ths integral 

(8) is improper and its finiteness and continuity must be examined. 
About the point (x', t/', z 1 } on S let us circumscribe a circle of radius a. 
If the radius is sufficiently small, the circular disk thus defined may be 
assumed plane. Now the potential at (x', ?/', z 1 ) due to surface charges 
outside the disk is a bounded and continuous function of position in the 
vicinity of (x', 2/', z')- Call this portion of the potential <fo. There 
remains a contribution fa due to the charge on the disk itself. The 



188 THE ELECTROSTATIC FIELD [CHAP. Ill 

surface density w is bounded, and we proceed as in Sec. 3.7. There exists 
a number m such that at every point on the disk |co| < ra, |co/r| < m/r. 



/n\ 1^1 * ^ 

(9) 0i = -: I - da 

v ' ' ' 



f * 1 j ma 
- I -rdr = ^ > 

Jo r 2e 



where $1 indicates that the surface integral is to be extended over the 
disk. The resultant potential at an arbitrary point (#', t/', z') on the 
surface is, hence, 

(10) = 0i + 2 . 

Both 0i and 2 have been shown to be bounded, and 2 is also continuous. 
But 0i vanishes with the area of the disk and consequently, since 
differs as little as desired from a continuous function, it is itself con- 
tinuous. The specialization of the disk to circular form is no restriction 
on the generality of the proof, since the circle may be considered to be 
circumscribed about a disk of arbitrary shape. The potential due to 
a surface distribution of charge is a bounded, continuous function of position 
at all points, both on and off the surface. The function defined by (8), 
therefore, passes continuously through the surface. 
The integral expression of the field intensity, 



(11) E(*', y', *') = 4 s <*, V, ) 

is continuous and has continuous derivatives of all orders at points not 
on the surface, but suffers an abrupt change as the point (x', y r , z') 
passes through S. The nature of the discontinuity may be determined 
directly from (II), 1 but we shall content ourselves here with the simple 
method employed in Sec. 1.13 based on the divergence and rotational 
properties of E. The transition of the vector E through the layer S is 
subject to a discontinuity defined by 

(12) E+ - E- = - con, 

^0 

where n is the unit normal drawn outward from the positive face of the 
surface. If now by o> we understand the true plus the bound charge, 
w n (P+ P_), then (12) is equivalent to 

(13) (e E+ + P+) n - (*_ + P_) n = (D+ - D_) n = . 

This specifies at the same time the transition of E at a surface of dis- 
continuity in a dipole distribution. 

3.16. Double-layer Distributions. It frequently happens that the 
potential of a charge distribution is identical with that which might be 

1 Cf. KELLOGG, loc. cit.; PHILLIPS, loc. cit., Chap VI. 



SBC. 3.16] DOUBLE-LAYER DISTRIBUTIONS 189 

produced by a layer of dipoles distributed over a surface. We imagine 
such a surface distribution to be generated by spreading positive charge 
of density +o> over the positive side of a regular surface S, and an identical 
distribution of opposite sign on its negative side. The result is a double 
layer of charge separated by the infinitesimal distance 1. The dipole 
moment per unit area, or surface density T, is a vector directed along the 
positive normal to S and is defined as the limit 

(14) T = n lim (wZ) as I > 0, co oo . 

The dipole moment corresponding to an element of area da on the double 
layer is r da, and its contribution to the poten- 
tial at a fixed point (x', y' } z') not on the sur- 
face is 

/^e\ jj 1 T COS , 1 _ /1\ 7 

(15) a< = -; 5 da = T t V I - I da. 

47T r 2 47T \r/ 

XT cos , 

.Now ~- " a measures exactly the solid angle 

d!2 at the point of observation ($', ^/', 2') sub- 
tended by the element of area da. This angle 
is positive if the radius vector r drawn from area Fl ^ ^Ue^dV 
(x f , y', z'} to the element da makes an acute solid angle at PI and a 
angle with the positive normal n. Thus in Fig. Iiegative angle at Pz ' 
33 the element da subtends a positive angle at PI and a negative one atP 2 . 
The potential due to the entire distribution may be written 

(16) <K*', y f f z') = -^ 




where is the solid angle subtended at (#', y', z'} by the surface S. That 
the second integral must be preceded by a negative sign is evident, if one 
notes that dtt is positive when the layer is viewed from the lower or 
negative side, where the potential is certainly negative. 

The potential <t> has distinct values on opposite sides of a double layer. 
Suppose first that the surface S is closed and that the distribution is of 
constant density, so that T may be taken outside the integral sign. The 
positive layer lies on the outer side of S so that T has the same direction 
as the positive normal. There is a well-known theorem on solid angles 
which applies to this case : " If S forms the complete boundary of a three- 
dimensional region, the total solid angle subtended by S at P is zero, 
if P lies outside the region, and 4?r if P lies inside." 1 The potential 

at any interior point of S is, therefore, 0- == r; at any exterior point 



1 On the analytic properties of solid angles see Phillips, loc. cit.,p. 112. 



190 THE ELECTROSTATIC FIELD [CHAP. Ill 

<+ = 0. The difference in potential on either side of the double layer is 

(17) <+ - <t>- = - r. 



Next we observe that (17) represents correctly the discontinuity in < 
as one traverses the double layer also when S is an open surface. For S 
may be closed by adding an arbitrary surface S f . At every point within 
or without this closed surface the resultant potential can be resolved into 
two parts, a fraction <t> due to the distribution on S, and a fraction <', due 
to that on S'. Inside the closed surface the resultant potential is again 

T; outside, it vanishes. Thus, as we traverse the surface S, <j> + <' 

60 

changes discontinuously by an amount T; but <', due to the layer S' y 



is certainly continuous across S and hence the entire discontinuity is in <, 
as specified by (17). 

It remains to show that (17) is also valid when r is a function of 
position on S. About an arbitrary point P on the surface S draw a circle 
of radius a. Let the radius a be so small that, over the area enclosed by 
the circle, r may be assumed to have a constant value TO. The potential 
()> in the neighborhood of P may again be resolved into two parts, a 
fraction <t>' due to the infinitesimal circular disk and a fraction <" due 
to that portion of the dipole layer lying outside the circle. <t>" is con- 
tinuous at P. </>' on the other hand suffers a discontinuous jump of T O 

CD 

on crossing the circular disk. The resultant potential < = $' + <t>" 
must, therefore, exhibit a discontinuity of the same amount and, in the 
limit as a > 0, we find that (17) holds for variable distributions if for T we 
take the value at the point of transition through S. 

These results might be interpreted to mean that the work done in 
moving a unit positive charge around a closed path which passes once 
through a surface bearing a uniform dipole distribution of density T O is 

TO, depending on the direction of circuitation. The potential in 



presence of such a double layer is a multivalued function; for to its value 

at (x' y y f j z') one may add mr, where m is any positive or negative 

o 

integer. To obtain a different value of < one need only return to the 
initial point after traversing the surface S. All this appears to be in 
contradiction to the conservative nature of the electrostatic field. As a 
matter of fact, the mathematical double layer constitutes a singular 
surface which has no true counterpart in nature. We shall show below 
by means of Green's theorem that the potential within a closed domain 



SEC. 3.16] DOUBLE-LAYER DISTRIBUTIONS 191 

bounded by a surface $ due to external charges is identical with that 
which might be produced by a certain dipole distribution over S. This 
equivalent double layer, however, does not lead to correct values of <t> 
outside S. 

The application of the integral (p E nda = - a to a small right 

J CQ 

cylinder, the ends of which lie on either side of the double layer, after the 
manner of Sec. 1.13, indicates at once that there is no discontinuity in 
the normal component of the field vector E across the surface since the 
total charge within the cylinder is now zero. 

(18) (E + - E_) - n = 0. 

The line integral of E around a closed path, however, is no longer neces- 
sarily zero; consequently we anticipate a possible discontinuity in the 
tangential components. Let the 
contour start at 1, Fig. 34, at which 
point the dipole density is r. The 
difference in potential between 
points 1 and 2 is, according to (17), 

J FIG. 34. Transition of the tangential com- 

</>! $2 = T. (Note that the ponents of E across a double layer. 



intensity E is given by the derivative of the potential, not the difference in 
potential across the discontinuity. The normal derivative is continuous.) 
If Al is the length of the path tangential to the surface, the density at 3 is 
r + Vr Al, and the potential difference between 4 and 3 is 

<* - 4>3 = - (r + Vr - Al). 
*o 

Now the quantity (< 2 <i) + (</>s $2) + (</>4 $3) + (<i $4) is 
identically zero; hence, 

(19) --T + (*s -*)+- (r + Vr - Al) + (4>! - $ 4 ) = 0. 

Q ^O 

Abbreviating < 4 <i = A<+, 4> 3 - fa = A(_, (19) reduces to 




where t is a unit vector tangent to S. The limit of A</AZ as Z 
is the component of E in the direction of t, so that in virtue of the con- 
tinuity of the normal component of E the transition of the field vector is 
specified by 

(21) E+ - E_ = VT. 

60 



192 THE ELECTROSTATIC FIELD [CHAP. Ill 

Clearly Vr signifies here the gradient of r in the surface and is, therefore, 
a vector tangent to $. The proof is subject to the assumption that r and 
its first derivatives are continuous over S. 

3.17. Interpretation of Green's Theorem. In Sec. 3.4 it was shown 
that the potential at any interior point of a region V bounded by a closed, 
regular surface S could be expressed in the form 

(22) *(*', y', z>) = * f p -dv + fl**, * {+*(?) da. 

v ' ^^ ' * ' ' 4njvr ^irjsrdn 4?r Js dn\r/ 

From the analysis of the preceding paragraph we are led to interpret this 
result in the following way. The volume integral represents, of course, 
the contribution of the charge within S, and the surface integrals account 
for all charges exterior to it. However, the first of these surface integrals 
is also equivalent to the potential of a single layer of charge distributed 
over S with a density 

(23) -. 

and the second can evidently be interpreted as the potential of a double 
layer on S whose density is 

(24) r = <. 

The charges outside S may be replaced by an equivalent single and double 
layer ', the densities of which are specified by (23) and (24), without modifying 
in any way the potential at an interior point. The potential outside S 
produced by these surface distributions corresponds in no way, however, 
to that arising from the true distribution. On the contrary, it is most 
important to note that these equivalent surface layers, which give rise to 
the proper value of the potential at all interior points of /S, are just those 
required to reduce both potential and field E to zero at every point outside. 
We observe first that the single layer gives rise to a discontinuity in the 
normal derivative of < equal to 



The normal derivative in Eq. (23) must be calculated on the inner or 
negative side of S, since this alone belongs to V. Replacing (d</dn)_ 
by its value (23), it follows that (d<f>/dri)+ = 0. Furthermore, the dis- 
continuity in potential due to the double layer is specified by (17), which 
together with (24) shows that <+ = 0. That <t> and its derivatives vanish 
everywhere outside S is readily shown by applying (22) to the volume F 2 
exterior to S. Since there are now no charges in F 2 , V 2 < = 0. At 
infinity the potential is regular, and consequently the potential within 



SBC. 3.18] 



IMAGES 



193 



7 2 is determined solely by the values of 0+ and (d<t>/dri)+ on S. But 
these have just been shown to vanish. The function <(#', y f , z f ) of (22) 
is continuous and has continuous derivatives. The field intensity E 
outside S is, therefore, also zero. 

From the foregoing discussion it is clear that one may always close 
off any portion of an electrostatic field by a surface, reducing the field 
and potential outside to zero, and taking account of the effect of external 
charges on the field within by proper single- and double-layer distribu- 
tions on the bounding surface. It is instructive to consider these results 
from the standpoint of the field intensity E. The single layer introduces 
the proper discontinuity in the normal component E n , but does not affect 
the transition of the tangential component. The double layer, on the 
other hand, in no wise affects the transition of the normal component, but 
may be adjusted to introduce the pro- 
per discontinuity in E t , or according 
to (21), 



f*',y',z r 



(26) &- = jj' ^+ = 0, 

where I is any direction tangent to S. 
If in particular the surface S is an 
equipotential, n X E = 0, and no 
dipole distribution is necessary; the 
field inside S due to external charges 
can then be accounted for by a single 

r\ i 

layer of density w = e -- on the 

equipotential. 

3.18. Images. An important ap- 
plication of these principles is to be 
found in the theory of images. The 
equipotential surfaces of a pair of 
equal point charges, one positive and 
the other negative in a homogeneous 
dielectric of inductive capacity e, form a family of spheres whose centers 
lie along the line joining the charges. Let the surface S be repre- 
sented by any equipotential about q, Fig. 35. This surface thus 
divides all space into two distinct regions, and in view of the regularity 
of the potential at infinity, Green's theorem applies to both, S being the 
bounding surface in either case; i.e., either region may be taken as the 
"interior" of S. If, therefore, the charge q be removed, the field in 
the region occupied by +q is unmodified if a charge of density o> is spread 
over S as specified by (23). Inversely, if the charge +q be located as 




Fio. 35. Application of Green's theorem 
to the theory of images. 



194 THE ELECTROSTATIC FIELD [CHAP. Ill 

shown with respect to a conducting sphere, a surface charge o> will be 
induced upon it such that it becomes an equipotential. The contribution 
of this induced charge to the field outside the conductor S is now deter- 
mined most simply by replacing the surface distribution by the equivalent 
point charge q. The charge q is said to be the image of +q with 
respect to the given sphere. In case the equipotential surface is the 
median plane x' = 0, the calculation is extremely simple. The potential 
at any point (x' } y' } z') to the right of x f = is 



(27) <t>(x', y', z'} = JL [ 

47T L 



*' - a) 2 



+ a) 2 + ?/ /2 + z 

The normal derivative at x f = must be taken in the direction of the 
negative z'-axis, since the region on the right of x f = is to be considered 
the interior of the equipotential. 



Q - 2 (a* + y" + *'*)* 
hence by (23) the charge density is 

<*> = - 

where r 2 = a 2 + */' 2 + z' 2 . 1 

BOUNDARY-VALUE PROBLEMS 

3.19. Formulation of Electrostatic Problems. The analysis of the 
preceding sections enables one to calculate the potential at any point in 
an electrostatic field when the distribution of charge and polarization is 
completely specified. In practice, however, the problem is rarely so 
elementary. Ordinarily only certain external sources, or an applied 
field, are given from which the polarization of dielectrics and the surface 
charge distribution on conductors must be determined such as to satisfy 
the boundary conditions over surfaces of discontinuity. 

Among electrostatic problems of this type are to be recognized two 
classes : the homogeneous boundary- value problem and the inhomogeneous 
problem. To illustrate the first, consider an isolated conductor embedded 
in a dielectric. A charge is placed on the conductor and we wish to know 
its distribution over the surface and the potential of the conductor with 
respect to earth or to infinity. At all points outside the conductor the 

1 On the method of images, see JEANS, "Mathematical Theory of Electricity and 
Magnetism," 5th ed., Chap. VIII, Cambridge University Press; or MASON and 
WEAVER, "The Electromagnetic Field," pp. 109jf., University of Chicago Press, 
1929. 



SBC. 3.19] FORMULATION OF ELECTROSTATIC PROBLEMS 195 

potential must satisfy Laplace's equation. At infinity it must vanish 
(regularity), and over the surface of the conductor it must assume a 
constant value. We shall show that these conditions are sufficient to 
determine </> uniquely. The density of surface charge can then be deter- 
mined from the normal derivative of $, subject to the condition that 
Jo; da over the surface of the conductor must equal the total charge. 

An inhomogeneous problem is represented by the case of a dielectric 
or conducting body introduced into the fixed field of external sources. 
A charge is then induced on the surface of a conductor which will dis- 
tribute itself in such a manner that the resultant potential is constant over 
its surface. The integral Jco da is now zero. Likewise there will be 
induced in the dielectrics a polarization whose field is superposed on the 
primary field to give a resultant field satisfying the boundary conditions. 

A schedule can be drawn up of the conditions which must be satisfied 
in every boundary-value problem. To simplify matters we shall assume 
henceforth that the dielectrics are isotropic and homogeneous except 
across a finite number of surfaces of discontinuity. 

(1) V 2 < = at all points not on a boundary surface or within external 
sources; 

(2) $ is continuous everywhere, including boundaries of dielectrics or of 
conductors, but excluding surfaces bearing a double layer; 

(3) <j> is finite everywhere, except at external point charges introduced as 
primary sources; 

(4) 2 ( -r- J ci ( J =0 across a surface bounding two dielectrics; 

(5) e ~ = co at the interface of a conductor and dielectric; 

tin 

(6) On the surface of a conductor either 

(a) <t> is a known constant <t>i, or 

(b) <t> is an unknown constant and 



(7) < is regular at infinity provided all sources are within a finite 
distance of the origin. 

In (4) it is assumed that the interface of the dielectrics bears no charge, 
as is almost invariably the case. The normal is directed from (1) to (2), 
and in (5) from conductor into dielectric. 

An electrostatic problem consists in finding among all possible solu- 
tions of Laplace's equation the particular one that will satisfy the con- 
ditions of the above schedule over the surfaces of specified conductors 
and dielectrics. 



196 THE ELECTROSTATIC FIELD [CHAP. Ill 

3.20. Uniqueness of Solution. Let < be a function which is harmonic 
(satisfies Laplace's equation) and which has continuous first and second 
derivatives throughout a region V and over its bounding surface S. 
According to Green's first identity, (5), page 165, <f> satisfies 



(1) I (V0) 2 dt> = I da. 

Suppose now that = on the surface S. Then f (V0) 2 dv = 0. The 

J v 

integrand is an essentially positive quantity and consequently V0 must 
vanish throughout V. This is possible only if is constant. Over the 
boundary = and, since by hypothesis is continuous throughout F, 
it follows that = over the entire region. 

Now let 0i and 02 be two functions that are harmonic throughout the 
closed region V and let 

(2) = 01 02. 

Then if 0i and 2 are equal over the boundary S, their difference vanishes 
identically throughout V. A function which is harmonic and which 
possesses continuous first- and second-order derivatives in a closed^ regular 
region V is uniquely determined by its values on the boundary S. 

Consider next a system of conductors embedded in a homogeneous 
dielectric whose potentials are specified. We wish to show that the 
potential at every point in space is thereby uniquely determined. The 
reasoning of the preceding paragraph is applied to a volume V which 
is bounded interiorly by the surfaces of the conductors, and on the 
exterior by a sphere of very large radius R. Let us suppose that there are 
two solutions, 0i and 2 , that satisfy the prescribed boundary conditions. 
Then on the surfaces of the conductors, = 0i 2 = 0. Since 0i 
and 02 are assumed to be solutions of the problem, they must both satisfy 
the conditions of Sec. 3.19; hence their difference is harmonic, has the 
value zero over the conductors, and is regular at infinity. The surface 
integral on the right-hand side of (1) must now be extended over both 
the interior and exterior boundaries. Over the interior boundary = 0; 
hence the integral vanishes. On the outer sphere d<j>/dn = 6<t>/dR. If 

R approaches infinity, vanishes as 1/R and d<t>/dR as 1/R 2 . The 
\ i 

integrand thus vanishes as l/# 3 , whereas the area of the sphere 

becomes infinite as R z . The surface integral over the exterior boundary 
is, therefore, zero in the limit R > oo . It follows too that the volume 
integral in (1) must vanish when extended over the entire space exterior 
to the conductors, and we conclude as before that if the two functions 0i 



SBC. 3.21] SOLUTION OF LAPLACE'S EQUATION 197 

and 02 are identical on the boundaries they are, identical everywhere: 
there is only one potential function that assumes the specified constant 
values over a given set of conductors. 

The left-hand side of (1) may also be made to vanish by specifying 
that d<p/dn shall be zero over the enclosing boundary S. Then through- 
out V we have again V< = 0, whence it follows that <f> is constant every- 
where, although not necessarily zero since the condition d<t>/dn = does 
not imply the vanishing of < on S. One concludes, as above, that if the 
normal derivatives d<t>i/dn and dfa/dn of two solutions are identical on the 
boundaries, the solutions themselves can differ only by a constant. In 
other words, the potential is uniquely determined except for an additive 
constant by the values of the normal derivative on the boundaries. But the 
normal derivative of the potential is in turn proportional to the surface 
charge density and, consequently, there is only one solution corresponding 
to a given set of charges on the conductors. 

In case there are dielectric bodies present in the field the requirement 
that the first derivatives of <, as well as <t> itself, shall be continuous is no 
longer satisfied and (1) cannot be applied directly. However, the region 
outside the conductors may be resolved into partial volumes V* bounded 
by the surfaces Si within which the dielectric is homogeneous. To each 
of these regions in turn, (1) is then applied. The potential is continuous 
across any surface Si and the derivatives on one side of Si are fixed in 
terms of the derivatives on the other. It is easy to see that also in this 
more general case the electrostatic problem is completely determined by the 
values either of the potentials or of the charges specified on the conductors of 
the system. 

3.21. Solution of Laplace's Equation. It should be apparent that 
the fundamental task in solving an electrostatic problem is the determina- 
tion of a solution of Laplace's equation in a form that will enable one to 
satisfy the boundary conditions by adjusting arbitrary constants. There 
are a certain number of special methods, such as the method of images, 
which can sometimes be applied for this purpose. Apart from the theory 
of integral equations, the only procedure that is both practical and 
general in character is the method known as " separation of the variables." 
Let us suppose that the surface S bounding a conductor or dielectric body 
satisfies the equation 

(3) /i(*, y, z) = C. 

We now introduce a set of orthogonal, curvilinear coordinates w 1 , w 2 , w 1 , 
as in Sec. 1.14, such that one coordinate surface, say u 1 = C, coincides 
with the prescribed boundary (3). If then a harmonic function </>(ti*, 
u 2 , u 3 ) can be found in this coordinate system, it is evident that the normal 
derivative at any point on the boundary is proportional to the derivative 



198 THE ELECTROSTATIC FIELD [CHAP. Ill 

of <j> with respect to u l , and that the derivatives with respect to u 2 and u* 
are tangential. 

According to (82), page 49, Laplace's equation in curvilinear coordi- 
nates is 



Let us suppose that the scale factors hi satisfy the condition 

(5) 

Each of the product functions /(w*) depends on the variable w* alone, 
but the factor M v does not contain u\ It may, however, depend on the 
other two. We next assume that < likewise may be expressed as a prod- 
uct of three functions of one variable each. 

(6) = FiWzMW). 

Then (4) can be written in the form 



If finally the Mi are rational functions, Eq. (7) can be resolved into three 
ordinary differential equations. The method is best described by 
example. 

1. Cylindrical Coordinates. From (1), page 51, we have 

r Vg VJ7 _ 1 Vg 

r, ^- - r, ^ - -p ^ - r, 

whence by inspection 

(9) /i = r, / 2 = / 3 = 1, M i = M 3 = 1, M 2 = 1 

Equation (7) becomes 



The first two terms of (10) do not contain z; the last term is independent of 
r and 4>. A change in z cannot affect the first two terms and therefore 
the last term must be constant if (10) is to be satisfied identically for any 
range of z. 



SEC. 3.21] SOLUTION OF LAPLACE'S EQUATION 199 

The arbitrary constant Ci, called the separation constant, has been chosen 
negative purely as a matter of convenience and the partial derivative 
has been changed to a total derivative since F 3 is a function of z alone. 

Upon replacing the third term of (10) by Ci and multiplying by r 2 
one obtains 

(12) L 
^ } Fi 

It is again apparent that the second term of (12) is constant, leading to 
two ordinary equations, 

(13) ~~ 2 + C*F* = 0, 



The equations for F 2 and F 3 are satisfied by exponential functions of 
real or imaginary argument depending on the algebraic sign attributed 
to the separation constants ; Fi is a Bessel function. In the not uncommon 
case of a potential < which is independent of 2, we find f\ constant, 
Ci = 0, and in place of (14) 



2. Spherical Coordinates. From (2), page 52, we have 



(16) V^ = r'sin*, =*sin*, = sin 0, 

whence 

(17) A = r, /, = sin e, /, = 1, 



T L T L gm 2 ( 

Equation (7) reduces to 

JL. 1 L**L \ + - l 1 / sin e ^\ + _ JL_ 
Fir 2 dr\ dr ) ^ r 2 sin OF 2 dO\ dO ) ^ r 2 sin 2 OF 



'3 VY 

where ^ has in this case been employed in place of <t> to represent the 
azimuthal angle. Separation of the variables leads to the three ordinary 
equations 



/IOTA sin 5 d / . d/' 1 2 /-, /-//, 

(196) m ~ = 2 ~ Cl ' 



= 0. 



Of these three, only the Legendre equation (196) is of any complexity. 



200 THE ELECTROSTATIC FIELD [CHAP. Ill 

3. Elliptic Coordinates. According to (3), page 52, we have 

(20) 



. /EE3, v_ /L3, 
\i-ii*' h\ ~V* 2 -i 

which by inspection lead to 



(21) 



_ 




The Laplace equation for the elliptic cylinder is 



(22) 



which upon separation gives 
(23a) 



(236) 

(28c) 

Both Fi and F t are Mathieu functions, but simplify notably when Ci 
as is the case when 4> is uniform along the length of the cylinder. 
4. Spheroidal Coordinates. According to (6), page 56, we have 

If ~ ~ ' ~W ~ c( ~ n >' ~h\~ 
/i = ? - 1, /i = l- u', /, = 1, 

t f C 1 f Ct -m f 




= 



(25) 



Laplace's equation reduces to 



1 = 0. 



which separates into the three ordinary equations 

(27o) kw = ~ Ci> 

(276) i^ 

(27c) i ^ 



SEC. 3.22] CONDUCTING SPHERE IN FIELD 201 

Equations (276) and (27c), which obviously are identical, are satisfied 
by the associated Legendre functions. 

The criteria stated on page 198 for the separability of Laplace's 
equation are not the most general known, but they include all coordinate 
systems of common use. A more general set of separable coordinate 
surfaces, of which the toroidal surfaces formed by rotating a set of bipolar 
coordinates is an example, has been investigated by Bdcher. 1 

It would lead us too far afield to discuss the properties of the various 
functions satisfying the equations enumerated above and the methods 
of determining the arbitrary constants. We shall have constant occasion 
throughout the remainder of this volume to employ Legendre and Bessel 
functions. The reader is referred to the classic treatises of Hobson 2 and 
Watson 8 , and to Whittaker and Watson. 4 Detailed application of 
boundary-value methods to electrostatics will be found in Jeans 5 and in 
Smythe. 6 

PROBLEM OF THE SPHERE 

3.22. Conducting Sphere in Field of a Point Charge. Consider a 
conducting sphere of radius TI whose center is located at the origin of the 
coordinate system. The sphere is embed- 
ded in a homogeneous, isotropic dielectric 
of inductive capacity 2 . At z = f > r\ 
on the z-axis, there is located a point 
charge q, Fig. 36. We wish to find the 
potential and the distribution of charge 
on the sphere. 

Let <t>o be the potential of the source q 
and <t>i the potential of the induced charge 
distribution on the sphere. The resultant 
potential at any point outside the sphere 
is < = <f> + <i. The induced potential $1 
must be single-valued, a condition satisfied FIG. 36. Sphere in the field of a point 

! i , , , 4. . n charge located at z f . 

only when the separation constant C 2 , 

Eq. (19c), page 199, is the square of an integer: C 2 = m 2 , m 0, 1, 
2, .... Likewise the only solutions of (196) which are finite and 
single-valued over the sphere are the associated Legendre function 

1 "Ueber die Reihenentwickelungen der Potentialtheorie," Dissertation, 1894. 

2 HOBSON, " Spherical and Ellipsoidal Harmonics," Cambridge University Press, 
1931. 

3 WATSON, "Treatise on the Theory of Bessel Functions," Cambridge University 
Press, 1922. 

4 WHITTAKBB and WATSON, "Modern Analysis," Cambridge University Press, 
1922. 

8 JEANS, loc. cit. 

6 SMYTHE, "Static and Dynamic Electricity," McGraw-Hill, 1939. 




202 THE ELECTROSTATIC FIELD (CHAP. Ill 



s 0), imposing on Ci the value Ci = n(n + 1), n = 0, 1, 2, . . . . 
Under these circumstances (19a) is satisfied by either r n or r"*" 1 . The 
condition that the potential be single-valued is fulfilled by the function 



(1) fa = y y anmr n + Z p-( cos g)e***, 

iti\ r J 

where a nm , b nm are arbitrary constants. But <t>i must also be regular at 
infinity, which necessitates our placing a nm = 0. Furthermore, the 
primary potential <o is symmetric about the z-axis; consequently m = 
in this case. The potential of the induced distribution is, therefore, 
represented by the series 

(2) 0i = 

The expansion of the primary potential < in spherical coordinates 
was carried out in Sec. 3.8. When r < f , 







and the resultant potential on the surface r = ri is 

) = 2 [if (?)" + ^] p - (cos 9) = 

u ^ ' J 



Now <^) 8 is a constant, and since (4) must hold for all values of 0, it follows 
that the coefficients of P n (cos 6) must vanish for all values of n greater 
than zero. The coefficients b n are thus determined from the set of 
relations 



At any point outside the sphere 



To determine the charge density, we compute the normal derivative 
on the surface. 



SBC. 3.22] CONDUCTING SPHERE IN FIELD 203 

and the induced charge density is 



, 

The total charge on the sphere is 



(9) q l = T ur\ sin 6 dd d$. 

Now a well-known property of the Legendre functions is their orthogonality. 

(10) P(cos 0)P(cos 0) sin d0 = 0, n p* w. 

We may take m = 0, P (cos 0) = 1, and learn that P n (cos 0) vanishes 
when integrated from to v if n > 0. 

(11) * qi = -q~ 
The potential of the sphere is, therefore, 



qi representing an excess charge that has been placed on the isolated 
sphere. If the sphere is grounded, <t> 8 may be put equal to zero. 

It is interesting to observe that the potential <i of the induced dis- 
tribution at any point outside the sphere is that which would be produced 

3 

by a charge 47re 2 & = Ji, a dipolc moment 4x6 2 6i = g ^ etc., all located 

at the origin and oriented along the z-axis (Sec. 3.8). There is, however, 
another simple interpretation. The point z = f ', Fig. 36, where f f ' = rf, 
is said to be the inverse of z = f with respect to the sphere. The recip- 
rocal distance from this inverse point to the point of observation is, by 
(8), page 173, 

[ n Pn(cOS 0) 



rn+1 



Thus the resultant potential (6) may be written 

Outside the sphere the potential is that of a charge q at z = f , an ii 
ri 
f 



7*1 

charge q' = 5 located at the inverse point z = f ', a charge gi (which 



204 THE ELECTROSTATIC FIELD [CHAP. Ill 

is zero when the sphere is uncharged) located at the origin, and a charge 
q -~ at the origin which raises the potential of the floating sphere to the 

proper value in the external field. 

3.23. Dielectric Sphere in Field of a Point Charge. At any point 
outside the sphere, whose conductivity is zero and whose inductive 
capacity is i, the potential is 



The notation <t>+ will be employed to denote a potential or field at points 
outside, or on the positive side, of a closed surface. The expansion of $1 
in inverse powers of r does not hold within the sphere, for the potential 
must be finite everywhere. We resort, therefore, to an alternative solu- 
tion of Laplace's equation obtained from (1) by putting the coefficients 
&nm equal to zero. At any interior point the resultant potential is 

00 

(16) <t>- = ^ a nrP n (cos 0), (r < n). 

n = 

<- includes the contribution of the charge q as well as that of the induced 
polarization, for the singularity occasioned by this point charge lies 
outside the region to which (16) is confined. In the neighborhood of the 
surface, r < |f|, so that <t> Q can be expanded as in (3). Just outside the 
sphere, 



Across the surface 

(18) ++ = *-, 

A calculation of the coefficients from these boundary conditions leads to 

< 2n + 1 

+(tt+l)6 2 ' 
-i U 



The potential at any point outside the sphere is 

+ 1 q q z ~ l ^ n 

" 



l + (n 

ns 

while at an interior point 



SBC. 3.24] SPHERE IN A PARALLEL FIELD 205 

It is important to observe that a region of infinite inductive capacity 
behaves like an uncharged conductor. As ei becomes very large, it ^ill 
be noted that the first term in the series (20) corresponding to n = 
vanishes, so that in the limit one obtains (6) for the case qi = 0. 

3.24. Sphere in a Parallel Field. As the point source q recedes from 
the origin, the field in the proximity of the sphere becomes homogeneous 
and parallel. We shall consider the case of a sphere embedded in a 
dielectric of inductive capacity c 2 under the influence of a uniform, 
parallel, external field E directed along the positive z-axis. The primary 
potential is then 



(22) 4> = -Eoz = -E<>r cos = -# rPi(cos 0). 

Note that </> is no longer regular at infinity, for the source itself is infinitely 
remote. The potential outside the sphere due to either induced surface 
charge or polarization is again 

(23) ^ 





n=-0 



If the sphere is conducting, the resultant potential on its surface and 
throughout its interior is a constant <,. 



(24) *. = -Sor 1 P 1 (cos 9) + 

n-0 

< a is independent of 6; whence it follows that 

(25) 6 = ri<., 61 = r\E Ql b n = when n > 1. 

(26) <t>+ = -E,r cos 6 + E,r\ C -~ + <t>, T 

The charge density and the total charge are respectively 

(27) co = 3e 2 # cos + ~> qi = lirrierf,. 

The potential of the induced surface charge is, therefore, that of a dipole 
of moment p = ^^E^r\] a moment, in other words, which is proportional 
to the volume of the sphere. To this is added the potential of q\ in case 
the sphere is charged. 

If the sphere is a dielectric of inductive capacity ei, the potential at 
an interior point is of the form (16). To satisfy the boundary conditions 
(18), the coefficients must now be 



flo = b = 0, 



a n = 6 n = 0, when n > 1. 



206 THE ELECTROSTATIC FIELD [CHAP. Ill 

The resultant potential is then 

= - Er cos 9 + ^^ r\E Q -J-?, 

+ 2e2 r 

u r cos 0. 



(29) 1 



Within //ie sphere the field is parallel and uniform. 

(30) *--!?- -rV * 

O2 1 + Je 2 

The dielectric constant KI of the sphere may be either larger or smaller 
than K 2 . Thus the field within a spherical cavity excised from a homo- 
geneous dielectric /c 2 is 

(31) * 



__,. 

Next we note that the induced field outside is that of a dipole oriented 
along the z-axis whose moment is 

(32) p = 



Apparently even a spherical cavity behaves like a dipole. This effect 
may be readily accounted for by recalling that the walls of the cavity 
bear a bound charge of density a/ = n P 2 , where P 2 is the polarization 
of the external medium. 

In the case of a dielectric sphere in air, e 2 = e . The polarization of 
the sphere is then 

(33) P! = eoOd - 1)E~ - 3 ^=4 e E > 

KI -f- * 

and its dipole moment 

(34) p = | Trr??! = 4xrl ^~ 6 E . 

The energy of this polarized sphere in the external field is 

(35) Ui = -I ( Pi Eo dv = -27rr? ^^ * E % = -J P Eo. 

& jv KI ~t~ ^ <6 

The dielectric polarization modifies the field within the sphere. It 
will be convenient to express this modification directly in terms of PI. 
A depolarizing factor L is defined by 

(36) E- = Eo - LPi. 

From (30) and the relation P! = O(KI 1)E~, one calculates for a sphere 
in a parallel external field 



SBO. 3.25] FREE CHARGE ON A CONDUCTING ELLIPSOID 

j 



207 



(37) 



3e 



In the case of a sphere immersed in air * 2 = 1, L = - 




FIG. 37a. Conducting sphere in a 
parallel field. The external medium 
is air. 



Fio. 376.- 
parallel field, 
is air. 



-Dielectric sphere in a 
The external medium 



PROBLEM OF THE ELLIPSOID 

3.25. Free Charge on a Conducting Ellipsoid. In ellipsoidal coordi- 
nates Laplace's equation reduces by virtue of Eq. (135), page 59, to 



(i) o-r) 



* 



+ ({ - n)B f 



(*)-" 



The properties of the ellipsoidal harmonics that satisfy this equation 
have been extensively studied, but we shall construct here only certain 
elementary solutions that will prove sufficient for the problems in view. 

Consider first a conducting ellipsoid embedded in a homogeneous 
dielectric e 2 . The semiprincipal axes of the ellipsoid are a, 6, c. It 
carries a total charge q, and we assume initially that there is no external 
field. We wish to know the potential and the distribution of charge over 
the conducting surface. 

To solve this problem a potential function must be found which 
satisfies (1), which is regular at infinity, and which is constant over the 
given ellipsoid. Now is the parameter of a family of ellipsoids all 
confocal with the standard surface = whose axes have the specified 
values a, 6, c. The variables rj and f are the parameters of confocal 
hyperboloids and as such serve to measure position on any ellipsoid 
= constant. On the surface = 0, therefore, $ must be independent of 
i) and f . If we can find a function depending only on which satisfies (1) 
and behaves properly at infinity, it can be adjusted to represent the 
potential correctly at any point outside the ellipsoid = 0. 



208 THE ELECTROSTATIC FIELD [CHAP. HI 

Let us assume, then, that < = <(). Laplace's equation reduces to 

(2) ~i fa ||) = 0, 

which on integration leads to 
(3) 



where Ci is an arbitrary constant. The choice of the upper limit is such 
as to ensure the proper behavior at infinity. When becomes very 
large, R$ approaches 8 and 

(4) 4>^~' ({->>). 

v 

On the other hand, the equation of an ellipsoid can be written in the form 

r 2 -J7 2 ^2 

( 5} x 4- -V -L. z t 

W 2 2 ^ *~ *' 



If r 2 = x 2 + y 2 + z 2 is the distance from the origin to any point on the 
ellipsoid , it is apparent that as becomes very large r 2 and hence 
at great distances from the origin 

ff*\ . *j\j i 

(6) 4 ~ - - 

The solution (3) is, therefore, regular at infinity. Moreover (6) enables 
us to determine at once the value of Ci; for it has been shown that, 
whatever the distribution, the dominant term of the expansion at remote 
points is the potential of a point charge at the origin equal to the total 

charge of the distribution in this case q. Hence Ci = -> and the 
potential at any point is 



The equipotential surfaces are the ellipsoids { = constant. Equation 
(7) is an elliptic integral and its values have been tabulated. 1 

To obtain the normal derivative we must remember that distance 
along a curvilinear coordinate u l is measured not by du l but by hi du 1 
(Sec. 1.16). In ellipsoidal coordinates 



rtn 

(8) 



l See for example Jahnke-Emde, "Tables of Functions," 2d ed., Teubner, 1933L 



SBC. 3.26] CONDUCTING ELLIPSOID IN A PARALLEL FIELD 209 

The density of charge over the surface = is 





If now in the three equations (132), page 59, defining x, y, z in terms of 
, i), f , we put = 0, it may be easily verified that 



~ 

a 4 4 c 4 



Consequently, the charge density in rectangular coordinates is 



t. 



Several special cases are of interest. If two of the axes are equal, the 
body is a spheroid and (7) can be integrated in terms of elementary 
functions. Thus, if a = b > c, the spheroid is oblate and 

(13) <t> = 

When c 0, the spheroid degenerates into a circular disk. On the other 
hand, if a > b = c the spheroid is prolate and we find for the potential 





The eccentricity of a prolate spheroid is e = ,Jl - ] As e 1, 



the spheroid degenerates into a long thin rod. 

3.26. Conducting Ellipsoid in a Parallel Field. We assume first that 
a uniform, parallel field E is directed along the o>axis, and consequently 
along the major axis of the ellipsoid. The potential of the applied field is 

Hto * 

(15) o = 



the value of x in ellipsoidal coordinates being substituted from Eqs. 
(132), page 59. This primary potential is clearly a solution of Laplace's 
equation in the form of a product of three functions, 



It is not, however, regular at infinity. 



210 THE ELECTROSTATIC FIELD [CHAP. Ill 

Now if the boundary conditions are to be satisfied, the potential fa 
of the induced distribution must vary functionally over every surface 
of the family = constant in exactly the same manner as fa. It 
differs from fa in its regularity at infinity. We presume, therefore, that 
fa is a function of the form 

(17) 
where 

(is) F,(i) 

To find the equation satisfied by (?i() we need only substitute (17) and 
(18) into (1), obtaining as a result 



na\ jf i 4- r n 

(19) R, ^ (R, -^ J - ^-4 + 2j Gi = 0. 

Equation (19) is an ordinary equation of the second order and as such 
possesses two independent solutions. One of these we know already to be 
F l = -y/ -|- a 2 . There is a theorem 1 which states that if one solution 
of a second-order linear equation is known, an independent solution can 
be determined from it by integration. If 7/1 is a solution of 

(20) g + p(x) | + q(x)y - 0, 
then an independent solution j/ 2 is given by 

J e -Cp dx 
yT dx ' 

In the present instance 



hence, 

/oo\ c< ( t\ 

(26) CriU) 

The limits of integration are arbitrary, but G\( ) is easily shown to vanish 
properly at infinity if the upper limit is made infinite. The potential of 
the induced charge is, therefore, 

(24) fa = 0o 7 



/ . [ i 

J See for example Ince, "Ordinary Differential Equations," p. 122, Longmans. 
1927. 



SEC. 3.27] DIELECTRIC ELLIPSOID IN PARALLEL FIELD 211 

The constant C 2 is determined finally from the condition that on the 
ellipsoid = the potential is a constant </> 8 . 



At any external point the potential is 

. , , <t>* <fro C 

* = * + p dg L 

J U + 



As in the analogous problem of the sphere, the constant <, can be cal- 
culated in terms of the total charge on the ellipsoid. The integrals 
occurring in (26) are elliptic and of the second kind. 1 

In case the field is parallel to either of the two minor axes it is only 
necessary to replace the parameter a 2 above by Z> 2 or c 2 . Thus the 
potential about a conducting ellipsoid oriented arbitrarily with respect 
to the axis of a uniform, parallel field E can be found by resolving E 
into three components parallel to the principal axes of the ellipsoid and 
then superposing the resulting three solutions of the type (26). 

3.27. Dielectric Ellipsoid in Parallel Field. It is now a simple matter 
to calculate the perturbation of a uniform, parallel field due to a dielectric 
ellipsoid. We shall assume that the inductive capacity of the ellipsoid 
is i, and that it is embedded in a homogeneous medium whose inductive 
capacity is again 2 The applied E is directed arbitrarily with respect 
to the reference system and has the components EQ X , E^ EQ Z along the 
axes of the ellipsoid. 

Consider first the component field E$ x . Outside the ellipsoid the 
resultant potential must exhibit the same general functional behavior 
as in the preceding example, and will differ from it only in the value of 
the constant C* In this region, therefore, 



(27) 



[d 



The variable s has replaced under the integral to avoid confusion with 
the lower limit. 

The interior of the ellipsoid corresponds to the range c 2 < J < if 
a ^ 6 > c. In this region <t>~ must vary with 17 and f as determined by 
the function F^(rf)F^) and, since (19) has only two independent solutions, 
the dependence on must be represented by either Fi() or (?i() to 
satisfy Laplace's equation. But the function (?i() is infinite at = c 2 , 

1 See for example Whittaker and Watson, "Modern Analysis," 4th ed., pp. 512f., 
Cambridge University Press, 1927. 



212 THE ELECTROSTATIC FIELD [CHAP. Ill 

whereas Fi() is finite at all points within the surface = 0. Thus the 
potential within this region must have the form 

(28) <t>- = CsF 1 (QF 2 (7 ? )F 3 (r), 

where C 3 is an undetermined constant. 

The constants C 2 and C 3 are to be adjusted to satisfy the boundary 
conditions 

(29) *+ = *- 
The first of these leads to 



the second gives 
(31) 



Since tf> = E Qx x, one finds that the potential at any interior point of 
the ellipsoid is 



(32) 



and the field intensity is 



The components along the other two axes are found in an identical 
manner. The potential of the applied field is in total 
(34) 4>o = -E 0x x - E Qv y - E 0lt z, 

and the resultant potential at an interior point is 



i ^<c, ij.r ^A 

1 + (ei - 2 )Ai 1 + (e! - c 2 )A 2 



in which the constants A 2 and A 3 are defined by 
(36) A2= 



( S 
The remaining components of the perturbed field are 






SBC. 3.28] CAVITY DEFINITIONS OF E AND D 213 

We are led to a conclusion of great practical importance: If the applied 
field is initially uniform and parallel, the resultant field within the ellipsoid 
is also uniform and parallel, whatever the orientation of the axes. The vec- 
tor E~, however, is not in general parallel to E , for the constants AI, 
A 2 , A 3 are equal only in degenerate cases. 

Outside the ellipsoid the resultant potential associated with the 
primary component E Qx is given by (27), 

ds 

l T ' I 7 

(38) 




i 4- ^ c 6l "" 6 2 f 00 ds 

2 c 2 Jo ( + a)B t 



with corresponding potentials for the fields E^ and E 0g . If in (38) we 
let ci -> oo , we find that <j>+ reduces to (26) for the case 0. = 0. The 
potential outsid$ a body of infinite inductive capacity is the same as that 
outside a grounded conductor of the same shape. 

The induced polarization PI within the ellipsoid tends to decrease the 
applied field. The depolarizing factors LI, L 2 , L 3 are defined by the 
relations 

(39) E; - E, x - LiP lx , E; = E 0v - LJP ly , E; = #o, - LJ> lz . 

Putting Pi = ( Cl 6 ) E~ and introducing the components of E~ from 
(33) and (37) leads to 



Most commonly the medium outside the ellipsoid is free space, /c 2 = 1. 
In that case the depolarizing factors depend only on the form of the 
ellipsoid. 



L * - A* (3 - 1, 2, 3). 

The polarization may then be determined from the simple formula 

(42) 




with corresponding expressions for P v and P f . 

3.28. Cavity Definitions of E and D. The field inside an ellipsoidal 
cavity is given by (33) and (37) when i = c . There are two cases of 
considerable interest: that of a disk-like cavity whose plane is normal to 
the direction of the applied field, and that of a needle-shaped cavity 
parallel to the field. 



214 



THE ELECTROSTATIC FIELD 



[CHAP. Ill 



We shall consider first the applied field to be oriented along the s-axis. 
Tnen as c becomes very small the ellipsoid degenerates into the disk 
illustrated in Fig. 38a. There is no loss of generality in assuming the 
disk circular, a = 6, and in this case the elliptic integral A 3 reduces to a 
more elementary function. 



(43) A 3 = 



(a 2 - c 2 )s 



- tan- 1 




The limit of the product a 2 cA 3 as c > is 2. The field inside the cavity 
is purely transverse, and since KI = 1, 

(44) E- = * 2 Eo= -Do. 



Apart from the proportionality constant , the field intensity E" within 
the disk-shaped cavity is equal to the vector Do of the initial field in the 
dielectric. 

A 



\j 



FIG, 38a. Fia. 386. 

FIGS. 38a and 6. Illustrating the cavity definitions of the vectors E and D. The arrows 
indicate the direction of the applied field. 

In Fig. 386 the field is directed along the major axis of the prolate 
spheroid a > b = c. 



(45) 



ds 




where e = .Jl ^ is the eccentricity. As e > 1, the spheroid degen- 
erates into a long, needle-shaped cavity and the product db*Ai approaches 
zero. At any point inside this cavity E~ = E : the field intensity E" is 
exactly the same as that prevailing initially in the dielectric. 

These cavity definitions of the vectors E and D in a ponderable medium 
were introduced by Lord Kelvin. Obviously a direct measurement of 
E or D in terms of the force or torque exerted on a small test body is not 



SBC. 3.29] TORQUE EXERTED ON AN ELLIPSOID 215 

feasible within a solid dielectric, and if a cavity is excised the field within 
it will depend on the shape of the hole. Our calculations have shown, 
however, that the electric field intensity within the dielectric is exactly 
that which might be measured within the needle-shaped opening of 
Fig. 386, whereas the field measured within the flat slit of Fig. 38a differs 
from D within the dielectric only by a constant factor. 

3.29. Torque Exerted on an Ellipsoid. The intensity of an electro- 
static field may be measured by introducing a small test body of known 
shape and inductive capacity suspended by a torsion fiber and observing 
the torque. Inversely, if the intensity of the field is known, such an 
experiment may be employed to determine the inductive capacity or 
susceptibility of a sample of dielectric matter. In general the field and 
polarization throughout the interior of the probe are nonuniform and an 
accurate computation is difficult or impossible. On the other hand, the 
advantages of an ellipsoidal test body for these purposes are obvious. 
The polarization of the entire probe is constant and the applied torque 
depends essentially only on its volume and its inductive capacity. 

According to Eq. (49), page 113, the energy of a dielectric body in an 
external field is 



(46) 



^ 1 f 

Jv 



where as above E~ denotes the resultant field inside the body and E 
the initial field. If the body is ellipsoidal and E is homogeneous, we 
have by (33) and (37) : 



(47) 



abc iii 



and, since the volume of an ellipsoid is fyrdbc, 

(48) U = |^ra6c(e 2 - ci)ET B . 

This energy depends not only upon the intensity of the initial field but also 
upon the orientation of the principal axes with respect to the field. Let 
the vector &o represent a virtual angular displacement of the ellipsoid 
about its center and T the resultant torque exerted by the field. Both T 
and 5o> are axial vectors (page 67), and the components Sco*, 6co y , a. are 
the angles of rotation about the axes x, y, z, respectively. The work done 
in the course of such a rotation is 

(49) 8W = T &> T, 3co, + !T y a? v -f T, Sco,. 



216 



THE ELECTROSTATIC FIELD 



[CHAP. Ill 



This work must be compensated by a decrease in the potential energy [7. 
In virtue of the homogeneous, quadratic character of (47), we may write 

(50) 817 = ra6c( 2 - i)(B" E ) = ^abc(e 2 - ci)Er - 5E . 

The reference axes have been chosen to coincide with the principal 
axes of the ellipsoid. Relative to this system fixed in the body the 
variation of E corresponding to a virtual rotation 5o> is 

(51) SE = Eo X $<o, 
whence for the energy balance we obtain 

(52) 8U = irrafccfe - ci)E" (E X 6o>) 

= $7ra&c(e 2 - i)(E" X E ) &> = ~&W, 

and from the arbitrariness of the components of rotation it follows that 
the torque exerted by the field is 

(53) T = ^7ra6c(i - 2 )E~ X E . 
The components of this torque are 



(1 -L abC Kl "~ 
I -J. 
^ ^2 



I _L 

1 + -^ 




(54) T v = 



T, = 



A,- 




To investigate the stability of the ellipsoid we must determine the 
relative magnitudes of the constants Ai, A 2 , A 3 . In the first place, it is 
clear from their definition, (32) and (36), that all three are positive, 
whatever the values of a, 6, c. It is easy to show, furthermore, that the 
order of their relative magnitudes is the inverse of the order of the 
three parameters. That is, if a > b > c, AI < A 2 < A 3 . Next, one 
finds that the sum of the three integrals can be reduced to a simple integral 
when u = Rl is introduced as a new variable, giving 



(55) 



+ A, + A, - 



PROBLEMS 217 

whence from the essentially positive character of the constants it follows 
that 



< A, S - c , (j = 1, 2, 3), 

(57) 1 + ^^ A , >0 , 0-1,2,8). 

The denominators of (54) are, therefore, positive both when 1 > 2 and 
when 61 < 2 , and the direction of the components of torque is independent 
of the relative magnitudes of ei and c 2 . If the applied field is parallel to 
any of the three principal axes, all components of torque are zero, so 
that these constitute three directions of equilibrium. 

The stability of equilibrium depends on the direction of the torque 
and this, we see, depends solely on the sign of A/ A k , which in turn 
is determined by the relative magnitudes of the axes a, 6, c. Thus 
A A 2 is positive if b > c, negative when c > b. The components of 
torque are such as to rotate the longest axis into the direction of the field 
by the shortest route. An ellipsoid whose major axis is oriented along 
the applied field is in stable equilibrium; the equilibrium positions of the 
minor axes are unstable. 

Problems 

1. The coordinates , TJ, f are obtained from the rectangular coordinates x, y t z by 
the transformation 



where / is any analytic function of the complex variable x -f iy. Demonstrate the 
following properties of this transformation : 

a. The differential line element is ds 2 = W(d? + cfy 2 ) + df 2 , where h = 1/|/'|, 
/' denoting differentiation with respect to the variable x -f- iy, 

b. The system , 17, f is orthogonal; 

c. The transformation is conformal, so that every infinitesimal figure in the xy- 
plane is mapped as a geometrically similar figure on the ^ij-plane. 

Show that in these coordinates the Laplacian of a scalar function assumes 
the form 

i / a2< fr r 5Lr i i 



and find expressions for the divergence and curl of a vector. 

2. With reference to Problem 1, discuss the coordinates defined by the following 
transformations : 

(1) + i? - In (x + ty), 

(2) x + iy - a cosh ( + trj), 

(3) * + ty - ( + ti)', 



218 THE ELECTROSTATIC FIELD [CHAP. Ill 

(5) + irj ~ In * "I" y + g i 

re + iy a 

(6) a; -f- iy = ia cot 



2 

3. A set of ring or toroidal coordinates X, MI ^ are defined by the relations 



r cos ^, y = r sin 



cosh X + cos A* 
where 

_ sinh X 
cosh X 4- cos /i 

Show that the surfaces, ^ constant, are meridian planes through the z-axis, that 
the surfaces, X = constant, are the toruses whose meridian sections are the circles 

z 2 + z* - 2x coth X + 1 ~ 0, 

and that the surfaces, ju constant, are spheres whose meridian sections are the 
circles 

a; 2 + z 2 + 2z tan /x - 1 0. 

Show that the system is orthogonal apart from certain exceptional points, and find 
these points. Find the expression for the differential element of length and show that 



_ **' *\ d ( r *+\ \ * ( r d +\.i r a20 1 

r 3 L dx \ ax/ a/i\ a/*/ sinh 2 xa^ 2 J 

Is Laplace's equation separable in these coordinates? 

4. Let F(:c, y, z) = \ represent a family of surfaces such that F has continuous 
partial derivatives of first and second orders. Show that a necessary and sufficient 
condition that these surfaces may be equipotentials is 



where /(X) is a function of X only. Show that if this condition is fulfilled the potential 
is 



where c\ and c 2 are constants. 

5. Show that \f/ =* F(z -\- ix cos u + tt/ sin w) is a solution of Laplace's equation 

dV av av . 
ao? 2 ay 2 az 2 

in three dimensions for all values of the parameter u and for any analytic function F. 
Show further that any linear combination of 2n -f 1 independent particular solutions 
can be expressed by the integral 



I (z + t-s cos u + t'v sin u) n f n (u) du> 
tJ~* * 



PROBLEMS 219 

where f n (u) is a rational function of e* u , and finally that every solution of Laplace's 
equation which is analytic within some spherical domain can be exoressed as an 
integral of the form 



= J_ 



f(z + ix cos u + iy sin w, u) du. 



(See Whittaker and Watson, "Modern Analysis/ 7 Chap. XVIII.) 

6. Charge is distributed along an infinite straight line with a constant density of g 
coulombs/meter. Show that the field intensity at any point, whose distance from 
the line is r, is 



and that this field is the negative gradient of a potential function 

(* 2/) flln-t 

27T6 T 

where r is an arbitrary constant representing the radius of a cylinder on which <jf> 0. 
From these results show that if charge is distributed in a two-dimensional space 
with a density o>(o;, y) the potential at any point in the zy-plane is 



1 f , r 
I w In da t 
2 J r 



where r \^(x' x) 2 + (y f y) 2 and that <(#, 2/) satisfies 



Show furthermore that this potential function is not in general regular at infinity, 
but that if the total charge 'iero, so that / co da = 0, then r<j> is bounded as r oo . 

7. Charge is distributed: over an area of finite extent in a two-dimensional space 
with a density <>(, y). Express the potential at an external point x', y' as a power 
series in r, the distance from an arbitrary origin to the fixed point a/, y'. Show that 
the successive terms are the potentials of a series of two-dimensional multipoles. 

8. Let C be any closed contour in the xy-plane bounding an area S and let </> be a 
two-dimensional scalar potential function. By Green's theorem show that 



I InrWda- I ( In r - <f> In r ) <fo, 
JS JC\ dn dn / 



where u <j>(x', y') if a?', y' is an interior point, and u = if x', y' is exterior. In this 
formula 



and n is the normal drawn outward from the contour. 

9, A circle of radius a is drawn in a two-dimensional space. Position on the circle 
is specified by an angle 0'. The potential on the circle is a given function <>(a, #') 



220 THE ELECTROSTATIC FIELD [CHAP. Ill 

Show that at any point outside the circle whose polar coordinates are r, the potential 
is 



This integral is due to Poisson. 

10. An infinite line charge of density q per unit length runs parallel to a conducting 
cylinder. Show that the induced field is that of a properly located image. Calculate 
the charge distribution on the cylinder. 

11. The radii of two infinitely long conducting cylinders of circular cross section 
are respectively a and b and the distance between centers is c, with c > a + b. The 
external medium is a fluid whose inductive capacity is e. The potential difference 
between the cylinders is V volts. Obtain expressions for the charge density and the 
mechanical force exerted on one cylinder by the other per unit length. Solve by 
introducing bipolar coordinates into Laplace's equation and again by application of 
the method of images. 

12. An infinite dielectric cylinder is placed in a parallel, uniform field E which is 
normal to the axis. Calculate the induced dipole moment per unit length and the 
depolarizing factor L (p. 206). 

13. Two point charges are immersed in an infinite, homogeneous, dielectric fluid. 
Show that the coulomb force exerted by one charge on the other can be found by 
evaluating the integral 



EE(E- n) - -E z n da, 

[Eq. (72), p. 152] over any infinite plane intersecting the line which joins the two 
charges. 

14. Two fluid dielectrics whose inductive capacities are i and 2 meet at an infinite 
plane surface. Two charges qi and q 2 are located a distance a from the plane on oppo- 
site sides and the line joining them is normal to the plane. Calculate the forces acting 
on qi and #2 and account for the fact that they are unequal. 

15. A conducting sphere of radius r\, carrying a charge Q, is in the field of a point 
charge q of the same sign. Assume q <3C Q. Plot the force exerted by the sphere 
on q as a function of distance from the center and calculate the point at which the 
direction reverses. 

16. A charge q is located within a spherical, conducting shell of radius b at a point 
whose distance from the center is a, with a < b. Calculate the potential of the sphere, 
the charge density on its internal surface, and the force exerted on q. Assume the 
sphere to be insulated and uncharged. 

17. Find the distribution of charge giving rise to a two-dimensional field whose 
potential at any point in the zy-plane is 



^ ( tan- 1 -f tan- 1 ) 

2^0 \ y y / 



18. Show that the electrostatic potential <f> is uniquely determined by the values 
of either < or d<t>/dn on the surfaces of conductors embedded in an isotropic, inhomo- 
geneous dielectric. 

19. A set of n conducting bodies is embedded in a dielectric medium which is iso- 
tropic but not necessarily homogeneous. If the charges q\, qz, . . . , q n are placed on 
these conductors the corresponding potentials on the conductors are fa, fa, ... 9 



PROBLEMS 221 

n . Likewise the charges q{, q(, . . . , q n give rise to the potentials <#>j, <^J, . . . , <f>' n . 
Show that 



20. Find the charge in coulombs on a metal sphere of radius a meters necessary to 
raise the potential one volt with respect to infinity. 

21. An infinite plane surface divides two half spaces, the one occupied by a dielec- 
tric of inductive capacity c, the other by air. A charge q is located on the air side a 
distance a from the surface. Obtain expressions for the potential at points in the air 
and in the dielectric and show that the potential in either space can be represented in 
terms of images. 

What is the density co' of induced, or bound, charge on the surface of the dielectric? 
Calculate the force acting on q and the work done in withdrawing the charge to 
infinity. 

22. The radii of two metal spheres are respectively a and b and the separation of 
their centers is c, where c > a + 6. A charge q a is placed on the first, thus raising its 
potential to a specified value < a . The potential fa of the second sphere is maintained 
at zero by grounding. Obtain expressions for the charges q a and qb and their distri- 
bution over the spheres by building up a series of images at the inverse points of the 
spheres (p. 203). 

The method is due to Lord Kelvin. See the discussion by F. Noether in Riemann- 
Weber's " Differential und Integralgleichungen der Physik," Vol. II, p. 281, 1927. 

23. Two infinite, parallel, conducting planes coincide with the surfaces x = 0, 
x =s a. The planes are grounded. A charge q is located on the #-axis at the point 
x = c, where < c < a. Show that the density of induced charge on the plane 
x = Ois 

f -^ 

^r >r^| j 2na c 2na + c ) 

"fa V' = 4^ ^ 1[(2na - c) 2 + r 2 ]* ~ [(2na + c) 2 + r 2 }*) 
n =, oo ^ / 

where r is the distance from the origin to the point x, y on the plane. What is the 
density on the plane x = a? Show that the total charge on the two planes is q. 
(Kellogg.) 

24. Prove that the surface charge density at any point on a charged ellipsoidal 
conductor is proportional to the perpendicular distance from the center of the ellipsoid 
to the plane tangent to the ellipsoid at the point. 

25. Equation (14), p. 209, gives the potential of a charged prolate spheroid. Show 
that as the length > oo and the eccentricity 1, this function approaches the 
logarithmic potential characteristic of a two-dimensional space. Apply the theorem 
of Problem 24 to find the charge per unit length. 

26. Charge is distributed with constant density within a thin ellipsoidal shell 
bounded by the two similar concentric ellipsoidal surfaces whose principal axes are a, 

6, c and all H 1> 6 ( 1 -\ ) >c (l "1 )' Show that the field is zero at all internal 

points and obtain an expression for the potential at external points. 

27. The major axis of a conducting, prolate spheroid is 20 cm. in length and its 
eccentricity is e. The spheroid is suspended in air and carries a total charge of q 
coulombs. The maximum field intensity at the surface shall at no point exceed 
3 X 10 volts /meter. Plot the maximum value of charge that can be placed on the 
spheroid subject to this condition as a function of eccentricity. 



222 THE ELECTROSTATIC FIELD [CHAP. Ill 

88. The effect of lightning conductors has been studied by Larmor (Proc. Roy. 
Soc., A, 90, 312, 1914) through the following problem. A hemispheroidal rod projects 
above a flat conducting plane. The potential of the charged cloud above is uni- 
form in the neighborhood of the rod. Find a potential function </> which is constant 
over the spheroidal surface and over the plane. Plot the ratio of potential gradient 
at the tip of the rod to intensity of the applied field against eccentricity of the rod. 
Assume the length of the rod (semimajor axis) to be fixed and equal to one meter. 
Discuss the lateral influence of the rod on the field as a function of eccentricity. 

29. A very thin, circular, metal disk of radius a carries a charge q. Show that the 
density of charge per unit area is 

1 



where r is the radial distance from the center on the disk. 

Calculate the field intensity at the surface of the disk and plot volts per meter per 
coulomb against r/a. Assume the disk to be suspended in air. 

30. A spheroidal, dielectric body is suspended in a uniform electrostatic field. 
The length of the major axis of the spheroid is twice that of its minor axis, and the 
specific inductive capacity is 4. The external medium is air. For what orientation 
with respect to the axis of the applied field is the torque a maximum? Calculate the 
value of this maximum torque per unit volume of material, per volt of applied field 
intensity. 

31. A stationary current distribution is established in a medium which is isotropic 
but not necessarily homogeneous. Show that the medium will in general acquire a 
volume distribution of charge whose density is 

Te - eVcr) V<f>, 

where <r and c are respectively the conductivity and inductive capacity of the medium. 

32. Two homogeneous, isotropic media characterized by the constants i, <TI, and 
2, <T2, meet at a surface S. A stationary current crosses S from one medium to the 
other. If the angles made by a line of flow with the normal at the point of transition 
are <j>i and $2, show that each line is refracted according to the law 

0*2 cotan $2 = <TI cotan <i. 
Show also that a charge appears on S whose surface density is 

-lAj-n. 



33. In many practical current-distribution problems it can be assumed that a sys- 
tem of perfectly conducting electrodes is embedded in a poorly conducting medium 
which is either homogeneous or in which discontinuities occur only across specified 
surfaces. The electrodes are maintained at fixed potentials and either these potentials 
or the electrode currents are specified. Show that the current distribution is uniquely 
determined by a potential function < satisfying the following conditions: 

(a) V 2 < = at all ordinary points of the medium; 

(6) < = 0, a constant on the ith electrode; 

(c) At a surface of discontinuity in the medium the transition is governed by 



PROBLEMS 223 

(d) Regularity at infinity; 

(e) Condition (6) may be replaced by / = (D o- da, where v is the condue- 

/ Si dn 

tivity of the medium just outside the electrode whose bounding surface is Si 
and where / is the total current leaving this electrode. 

34. A system of electrodes is embedded in a conducting medium. The electrodes 
are maintained at the fixed potentials fa, fa, . . . , <t> n , and the currents leaving the 
electrodes are I\, /2, . . . , /n. Show that the total Joule heat generated in the 
medium is 

Q = 

36. Prove that for a system of stationary currents driven by electromotive forces 
the current will distribute itself in such a way that the heat generated will be less for 
the actual distribution than for any other which coincides with the actual distribution 
in the region occupied by the driving forces and which is solenoidal everywhere else. 
Assume that the medium is isotropic and that the relation between field intensity and 
current density is linear. 

36. A circular disk electrode of radius a is in contact with an infinite half space of 
conductivity <r, such as the surface of the earth. The distance to all other electrodes 
is large relative to the radius a. Show that the current distribution is determined by 
the conditions: 

(a) V 2 < within the conducting half space; 

(6) 3<f>/dn = at the boundary excluding the disk; 

(c) = -- . _____ ..... - on the disk, where r is the radial distance from the 
dn 2ir<ra V & r * 
center. 

37. A stationary current is carried by a curved, conducting sheet of uniform thick- 
ness. The sheet is assumed to be so thin that the current distribution is essentially 
two-dimensional. If and 17 are curvilinear coordinates on the surface, the geometry 
of the surface is defined by the element of arc 

ds* = 0n d? + 2^12 d drj + 022 <fy 2 , 

the properties of the g^ having been defined in Sec. 1.14. The scalar potential deter- 
mining the current distribution satisfies the Laplace equation 



-4- 



"~~ ' - 

d * V 011022 ~ 

Define a conjugate function ^ by the relation 

d<ft d<j> d<^> d<j> 

011 - -- 012 77 022 012 T- 

,, ft? _ d Jt . , d _ ftj , 

d$ - -- d$ + ' ^r dlj 

V0H022 ~ 0I 2 V 011022 - 0? 2 

and show that ^ satisfies the same equation as <f>. The functions <f> and $ are potential 
and streamline functions on the surface, and $ =* <t> + i\f/ will be called a complex 



224 THE ELECTROSTATIC FIELD [CHAP. Ill 

function on the surface. Now choose and t\ so that f = 4* ii? is itself such a func- 
tion. Show that in this case 4> satisfies the simple equation 



and that <f> and ^ are related by 



while the line element reduces to ds* = g\\(d$* -\- <V). Thus the complex func- 
tions of the surface are analytic functions of each other and the transformation 
-r- irj = x + iy maps the surface conform ally on to the complex z-plane. If the 
conformal mapping of a surface on to the plane is known, the current distribution 
problem reduces to the solution of Laplace's equation in rectangular coordinates. 
Inversely, if a current distribution over a curved surface can be determined experi- 
mentally, a conformal mapping of the surface on the plane is found. (Kirchhoff.) 
38. Referring to Problem 37, show that 

z = ae 1 ^ tan 
2 

is a conformal, stereographic projection of a sphere of radius a on the complex plane, 
where z = x -f- iy and p is the equatorial angle or longitude and the colatitude on 
the sphere. Show further that 

tan^-/() 



| j 



is a complex potential function on the sphere where / is any analytic function of z. 

39. A steady current I enters a conducting spherical shell of radius a and surface 
conductivity <r at a point on the surface and leaves at a similar point diametrically 
opposite. Find the potential at any point and the equation of the streamlines. 

40. A cylindrical condenser is formed of two concentric metal tubes. The space 
between the tubes is occupied by a solid dielectric of inductive capacity which, how- 
ever, is not in direct contact with the tubes, but separated from them by thin layers 
of a nonconducting intermediary fluid of inductive capacity c'. The solid dielectric 
extends so far beyond the ends of the metal tubes that the stray field is of negligible 
intensity. A constant potential difference is maintained between the metal elec- 
trodes. Obtain an expression for the change in length per unit Length of dielectric. 
(Kemble, Phys. Rev., 7, 614, 1916.) 



CHAPTER IV 

THE MAGNETOSTATIC FIELD 

The methods that have been developed for the analysis of electro- 
static fields apply largely to the magnetostatic field as well. Every 
magnetostatic field can be represented by an electrostatic field of identical 
structure produced by dipole distributions and fictive double layers. 
The equivalence, however, is purely formal. There is no quantity in 
magnetostatics corresponding to free charge, and the surface singularity 
generated by a double layer of electric charge does not actually exist. A 
double layer leads in fact to a multivalued potential and consequently to 
a nonconservative field. Whatever the analytical advantages of the 
electrostatic analogy may be, it is well to remember that the physical 
structure of a field due to stationary distributions of current differs 
fundamentally from that of any configuration of electric charges. 

GENERAL PROPERTIES OF A MAGNETOSTATIC FIELD 

4.1. Field Equations and the Vector Potential. The equations satis- 
fied by the magnetic vectors of a stationary field are obtained by placing 
the time derivatives in Maxwell's equations equal to zero. 

(I) V X H - J, (II) V B - 0. 

To these must be added the equation of continuity which now reduces to 
(III) V J = 0. 

The current distribution in a stationary field is solenoidal; all current lines 
either close upon themselves, or start and terminate at infinity. From (II) 
it follows likewise that all lines of the vector B lines of magnetic flux, as 
they are commonly called close upon themselves. Suppose that the 
flux density of the field produced by a current filament Ii is BI. All lines 
of this field link the circuit Ii. A fraction of the flux BI may, however, 
thread also a second current filament 7 2 . The concept of "flux linkages " 
commonly employed in the practical analysis of electromagnetic problems 
is based on these solenoidal properties of current and flux in the station- 
ary state. The extension to slowly varying fields (quasi-stationary 
state) is justifiable only so far as dp/dt remains negligible with respect 
to V J. 

Upon applying Stokes' theorem to (I), we obtain the equivalent 
integral equation, 

225 



226 THE MAGNETOSTATIC FIELD [CHAP. IV 

(la) f c H ds = f s J n da = /, 



where S is any surface spanning the contour C, and I is the total current 
through this surface. The line integral of H around a closed path is 
equal to the current linked : the magnetic field is nonconservative. 

Every solenoidal field admits a representation in terms of a vector 
potential. As in Sec. 1.9, Eq. (II) is identically satisfied by 

(1) B = V X A. 



The vector A is now chosen to satisfy (I), and the relation H = 
between the magnetic vectors must therefore be specified. If the 
medium is nonferromagnctic, the relation is linear; if moreover it is 
homogeneous and isotropic, we put B = juH and arrive at 

(2) V X V X A = /J 

for the vector potential. 

In Sec. 1.9, it was shown that there may be added to A the gradient 
of any scalar function \l/ without affecting the definition (1). By a proper 
choice of ^ it is always possible to effect the vanishing of the divergence 
of A. The vector potential is uniquely defined by (2) plus the condition 

(3) V . A = 0. 

In rectangular coordinates, (2) then reduces to 

(4) V 2 A - - M J, 

where it is understood that the Laplacian operates on each rectangular 
component of A. 

4.2. The Scalar Potential. The existence of a scalar potential func- 
tion associated with the electrostatic field is a direct consequence of the 
irrotational character of the field vector E. If double-layer distributions 
of charge are excluded, the curl of E is everywhere zero; hence one may 
express E as the negative gradient of a scalar <f>. Furthermore, since the 
line integral of E vanishes about every closed path of the region, the poten- 
tial is a single- valued function of the coordinates. 

A region of space, enclosed by boundaries, is said to be connected 
when it is possible to pass from any one point of the region to any other 
by an infinice number of paths, each of which lies wholly within the region. 
Any two paths which can by continuous deformation be made to coincide 
without ever passing out of the region are said to be mutually recon- 
cilable. Any closed curve or surface is said to be reducible if by con- 
tinuous variation it can be contracted to a point without passing out 
of the region. Two reconcilable paths when combined form a reducible 
circuit. If finally the region is such that all paths joining any two points 



SEC. 4.2] THE SCALAR POTENTIAL 227 

are reconcilable, or such that all circuits drawn within it are reducible, 
it is said to be simply connected. The electrostatic field of a volume 
distribution of charge constitutes a simply connected region of space 
within which every closed curve is reducible. 

Contrast this geometrical property of the electrostatic field structure 
with the magnetic field of a stationary distribution of current. We shall 
call the region of space occupied by current Vi, while the region within 
which J = will be denoted as 7 2 . Figure 39 represents a plane section 
through the field, the shaded areas indicating, for example, the cross 




FIQ. 39. Illustrating a doubly connected space. 

sections of conductors carrying current. In the region Vi the curl of H 
does not vanish and consequently no scalar potential exists. Throughout 
F 2 we find 

(5) V X H = 0, 

and so within this region H may be expressed as the negative gradient 
of a scalar function <*(#, y, z). 

(6) H = -V**. 

The line integral of H along any contour wholly within F 2 connecting 
two points P and Q is 

(7) H ds = - V<* ds = **(P) - **(Q). 



The paths by which one may reach Q from P are not, however, all mutu- 
ally reconcilable. A closed curve composed of two segments such as 
PEQ and QFP is irreducible, since on contraction to a point it must 
necessarily penetrate the region FI. Thus the region F 2 occupied by the 
magnetostatic field exterior to the current distribution is multiply 
connected. The scalar potential 0* is a multivalued function of position, 
for to its value at any point P one may add a factor nl by performing n 
complete circuitations about the current /. However F 2 may be rendered 
simply connected and the potential single-valued by the introduction 



228 THE MAGNETOSTATIC FIELD [CHAP. IV 

of a cut or barrier which prevents a circuit from being closed in such a 
manner as to link a current. There are, of course, an infinite number of 
ways in which such an imagined barrier might be placed but, if the cur- 
rent is constituted of a bundle of current filaments forming a single, 
closed current loop, it is most simple to think of it as a surface spanning 
the loop. The line MN in Fig. 39 represents the trace of such a sur- 
face. In the "cut" space the line integral of H about every closed path 
vanishes, for the barrier surface excludes current linkages, and con- 
sequently the potential function $* is single-valued. At points lying on 
either side and infinitely close to the barrier the values of <* differ by 
/. There is a discontinuity in the potential function thus defined equal 
to 

(8) *?-** = /, 

where the subscripts refer to the positive and negative sides of the surface. 
According to Sec. 3.16 the discontinuity in <* across the barrier is equivalent 
to that of a surface dipole layer of constant moment I per unit area. Only 
when all currents are zero and the sources of the field are permanent 
magnets does the potential become a single- valued function of position. 

4.3. Poisson's Analysis. At every point where the current density 
is zero, and in particular at interior points of a permanent magnet, we 
have 



(9) B = -MoVtf>* + M o(M + Mo). 

Since B is divergenceless, the scalar potential satisfies a Poisson equation 

(10) W* = -p*, p* = -V (M + Mo). 

The quantity p* is the magnetic analogue of "bound charge" density, 
which in the older literature was frequently called the density of Pois- 
son's "ideal magnetic matter." 

Across any surface of discontinuity in the medium the magnetic 
vectors satisfy, according to Sec. 1.13, the conditions 

(11) n (B 2 - Bi) =0, n X (H 2 - HI) = 0. 

The equivalent conditions imposed on the scalar potential are, therefore, 



Q2) - 

(12) dn A \dnt ' \dt 

(13) * = n [(M + M )i - (M + Mo) J. 

Within any closed region containing permanent magnets and polarizable 
matter , but throughout which the conduction current density J is zero, the 
magnetostatic problem is mathematically equivalent to an electrostatic 
problem. 



SBC. 4.3] POISSON'S ANALYSIS 229 

The calculation of the potential <* from the densities p* and w* has 
been described in the previous chapter. 



', y', z') J P *(x, y, 2) - cfo + ^ J *(, y, 2) i da, 



(14) 



where as usual r = \/(z' z) 2 + (y' y) 2 + (z 1 z) 2 . To evaluate 
the integrals one must know both the permanent magnetization M 
and the induced magnetization M; but the induced magnetization is a 
function of the field intensity and consequently of <* itself. If, however, 
the permanently magnetized bodies are not too close to one another, the 
induced magnetization can usually be neglected with respect to M . 
Some assumption is then made regarding the magnetization M and the 
fixed or primary field of the sources calculated by means of (14). This 
primary field induces a magnetization M in neighboring polarizable 
bodies soft iron, for example. The calculation of the induced or sec- 
ondary field is a boundary-value problem to be discussed later. 

Let us calculate now the magnetic moment of a magnetized piece of 
matter. By analogy with the electrostatic case, Sec. 3.11, this moment 
is defined by the integral 

(15) m = f v n P *(f , 17, ftdv + f s n *>*({, n, f ) da, 

where ri = i + fa + kf , Fig. 30, page 178, is the vector drawn from the 
origin to the element of ''magnetic charge/' and S the boundary of a body 
whose volume is 7. Upon replacing p* and o>* by their values defined 
in (10) and (13), then transforming by means of the identity 



V . M = V - (M) - M V, 
which applies to each component of TI, we find that (15) reduces to 

(16) m = f y (M + Mo) dv. 

The magnetic moment of a body is equal to the volume integral of its 
magnetization. 

Finally, if Eq. (14) is transformed by (9), page 184, after p* and * 
have been replaced by (10) and (13), one obtains for the potential 

(17) **(*', </', z'} - ~ J^ (M + Mo) - V (i) dv. 

If the dimensions of the magnetized body are sufficiently small with 
respect to the distance to the observer at (x', y', z'), the variation of 



230 THE MAONETOSTATIC FIELD [CHAP. IV 

Vl/r within V may be neglected and we are led to an expression for the 
scalar potential of a magnetic dipole, 

(18) <*(z' ; y', z') = j- m ' 

CALCULATION OF THE FIELD OF A CURRENT DISTRIBUTION 
4.4. The Biot-Savart Law. It has been shown that in a homogeneous, 
isotropic medium of constant permeability /* the vector potential satisfies 
the system of equations 



(1) W, = -W,, V A = 0, 0' = 1, 2, 3). 

The Aj are rectangular components of the vector A, but the restriction to 
rectangular coordinates does not necessarily apply to the scalar operator 
V 2 . Considering the Aj and *// as variant scalars, the theory developed 
in Sec. 3.4 for the integration of Poisson's equation may be applied directly 
to (1). If the current distribution can be circumscribed by a sphere of 
finite radius each component of the vector potential can be expressed 
as an integral, 



= ^ J 



(2) A y (*', y', z') = f- J&, y, z) dv, (j = 1, 2, 3), 

^tTT ^/ / 

extended over all space. These components are now recombined to 
give a vector. 

/o\ A //>.' /.' '^ ' 

\P) *i(x , y j z ; -r 



where r = \/(z' x) 2 + (y f y) 2 + (z f z) 2 . Moreover, it follows 
from the discussion of Sec. 3.7 that if J is a bounded, integrable function, 
then A is a continuous function of the coordinates (x* ', y' , z') possessing 
continuous first derivatives at every point inside and outside the current 
distribution. If J and all its derivatives of order less than n are finite 
and continuous, the vector function A has continuous derivatives of all 
orders less than n + 1. 

The vector potential defined by (3) satisfies the condition V A = 
provided the current distribution is spatially bounded. The integral 
is regular and consequently may be differentiated under the sign of 
integration. 

(4) V'.A = jj(*,3/, 2 ).V'(i)<fo= - J 

(5) J- 

\ * j 



SBC. 4.4] THE BIOT-SAVART LAW 231 

in virtue of the condition V J = 0. Then 



(6) 



--f f 

4irJ 



for, since all current lines close upon themselves within a finite domain, 
the surface S may be chosen such that there is no normal component 
of flow. 

Note that the contribution dA to the vector potential is directed 
parallel to the element of current J dv. Equation (3) can be simplified 
in the case of currents in linear conductors; i.e., conductors whose cross 
section da is so small with respect to the distance r that the density vector 
may be assumed uniform over the cross-sectional area and directed 
along the wire. 

(7) J dv = J da ds = I ds, 

where / is the t6tal current carried by the conductor and ds is an element 
of conductor length. Since the steady current 7 must be constant 
throughout the circuit, Eq. (3) reduces to 

(8) A(*', y', ') = I 

in which the integral is to be extended around the complete circuit C. 
The field can be obtained directly from (3) by calculating the curl of A 
at the point (x', y', z'). In virtue of the convergence of the integral and 
the continuity of its first derivatives the differential operator may be 
introduced under the sign of integration. 



(9) H(x' )2/ ' )2 ')= 

Expansion of the integrand according to elementary rules yields 

(10) V X J(x, y, z) = V X J(x, y, z) + V X J(x, y, z). 



But the current density is a function of the running coordinates x, y, z t 
whereas the differentiation applies only to x', y', z'\ consequently the first 
term on the right is zero, giving for the field vector 

dv 



(11) H(*', */', *') = ~ J V 0) X J(x, y, z) 



232 



THE MAGNETOSTATIC FIELD 



[CHAP. IV 



If r is a unit vector directed from the current element toward the point 
of observation, (11) is equivalent to 



(12) 



which for linear circuits assumes the form 



(13) 



I_ 

47T 



s X r 



ds, 



where s is a unit vector in the direction of the current element I ds. 
The law expressed by (13) is frequently formulated with the statement 
that each linear current ele.nent / ds contrib- 
utes an amount 



*~ 



* 




(14) 



s X r 



to the total field. This is essentially the result 
deduced from the experiments of Biot and Sav- 
art in 1820. There is no very cogent reason 
to reject the formula (14), other than the fact 
that this resolution of (13) into differential 
elements is not unique; for there may be added 
to (14) any vector function integrating to zero 
around a closed circuit. All stationary cur- 
rents are composed of closed filaments, so that 
it is difficult to imagine an experiment to deter- 
mine the contribution of an isolated element. 
Obviously no error can result from an 
application of (14) to the analysis of a field 
when it is understood that these contribu- 
tions are eventually to be summed around the 
circuit. 

Whereas the vector potential due to a differential element of current 
is directed parallel to the element, the field vector H is oriented normal 
to a plane containing both the current element and the line joining it 
with the point of observation; i.e., to the plane defined by the unit 
vectors s and r. If 6 is the angle which r makes with s, the intensity 
of the field is 



FlQ. 



40. Solenoidal current 
distribution. 



(15) 
v / 



'da. 



The field along the axis of a solenoid, Fig. 40, may, for example, be 
calculated approximately by assuming an equivalent current sheet of 



SBC. 4.5] EXPANSION OF THE VECTOR POTENTIAL 233 

negligible thickness. If there are n turns/meter carrying a current of 
J amp., the current density of the sheet is K = In. The contribution 
of a solenoidal element of radius a and length dx at a distance x along 
the axis from the point of observation is 

(16) dH = ~ T^rq-^p dx =* ^ d ( cos *) 

These contributions are directed along the axis. At any point on the 
axis, the field is 

cos 0i). 




At the center this reduces to 
(18) H = nl 



Another elementary case of some interest is that of a straight wire of 
infinite length and circular cross section. The current I is assumed to be 
distributed uniformly over the cross section. The simplest solution is 
obtained by taking advantage of the cylindrical symmetry of the field to 

apply the integral law (f>H ds = I. If r is the radial distance from the 
axis and a the radius of the wire, it is obvious that 

2*rH = TrrV = J, H = JL r, (r < a), 

(19) 

= 7, H = (r > a). 



The field outside the wire, which is independent of its radius, can also be 
obtained from the Biot-Savart law. If, however, one attempts to cal- 
culate the vector potential by means of (8), it will be noted that the 
integral diverges owing to the fact that the current distribution is in 
this case not confined to a region of finite extent. The vector potential is 
actually 

(20) A r = A, = 0, A, = ^ In , (r > a), 

as may be verified by calculating the curl in cylindrical coordinates 
(page 51), and this function becomes infinite as r > <*> . 

4.6. Expansion of the Vector Potential. Following the example set 
in Sec. 3.11 for the scalar potential, we shall determine an expansion of 
the vector potential of a stationary current distribution in terms of the 
coordinates of a point relative to a fixed origin. To ensure regularity at 



234 THE MAQNETOSTATIC FIELD [CHAP. IV 

infinity it will be assumed that the entire distribution can be circum- 
scribed by a sphere of finite radius R drawn from the origin; we shall 
consider here only points of observation exterior to this sphere. The 
current distribution is completely arbitrary, but it must be assumed that 
the permeability /* is everywhere constant and uniform. Practically 
this excludes only ferromagnetic materials and admits conductors of any 
other nature and arbitrary shape embedded in dielectrics of any sort. 

The notation will be essentially that adopted in Sec. 3.11 and illus- 
trated in Fig. 30; for convenience in summing we shall let the coordinates 
of the fixed point P be xi, x 2 , z 3 , and those of the current element 1, 2 , 3. 
The distance from the origin to P is r = -\/x\ + x\ + x\ and that from 
the element J( l; 2; 3 ) dv to P is 



At P the vector potential is, therefore, 



extended over the volume V bounded by the sphere of radius R. If ri 
is the vector whose components are /, the expansion of l/r 2 in a Taylor 
series about leads to 



Consequently the vector potential can be represented by the expansion 

(23) Ad,) = 2 AW = 5 7 J J.> * - 5 / [" T 0)] J<> * 



[r,.v(i)]} 



. vr,.v ,,*-.... 

Note that J and ri are functions of the / with respect to which the integra- 
tion is to be effected; the function 1/r depends only on the x 3 -, and the 
operator V applies to these coordinates alone. 

In virtue of its stationary character the current distribution may be 
resolved into filaments, all of which close upon themselves within V. 
Let us single out any one of these filaments. We suppose the infinitesimal 
cross section to be da\ and the current carried by this linear circuit to be 
I = |Jj dai. A point on the circuit is located by the vector TI drawn 



SEC. 4.5] 



EXPANSION OF THE VECTOR POTENTIAL 



233 



from the origin, Fig. 41, and an element of length along the circuit in 
the direction of the current is dri. Then for each filament or tube 



(24) 



A<> = i f dn = 0, 
4rrr J c ' 



where C is the closed filament contour. (A (0) would not necessarily 
vanish were V to contain only a portion of the current distribution.) 

The dominant term of the expan- 
sion is, therefore, the second, which 
we transform as follows. 

C^ 
i / 1 \ i i 

(25) 




Again the total differential vanishes FlG - 41. -Closed current filament follow- 

, ing the contour C. 

when integrated about a closed 

filament, while ri X dr\ = n da is the vector area of the infinitesimal 

triangle shown shaded in Fig. 41. 



(26) 

The quantity 

(27) 



A< = F 5 . 



m 



= / I n da = ~ I 



is by definition the magnetic dipole moment of the circuit. Since the 
surface integral may be extended over any regular surface spanning the 
contour, the magnetic moment depends only on the current and the form 
of the contour. 

The dipole potential of the total distribution is obtained by summation 
over all the current filaments. By definition the magnetic dipole moment 
of a volume distribution of current with respect to the origin is 



(28) 



m = H 



The magnetization M of a region carrying current can likewise be defined 
as the magnetic moment per unit volume. 



(29) 



M - 



dv 



- r, X 

~ 2 fl X 



It is clear that magnetization with respect to a given origin will occur 
whenever the current density has a component transverse to the vector 



236 THE MAGNETOSTATIC FIELD [CHAP. IV 

TI; that is to say, whenever the charge has a motion of rotation about 
as a center. If p is the charge per unit volume and v its effective velocity, 
so that J = pv, the magnetization is 

(30) M = in X pv, 

which is identical with angular momentum per unit volume when p is 
interpreted as the density of mass. 

The remaining terms of the expansion (23) are the vector potentials 
of magnetic multipoles of higher order. Two distinct definitions of 
magnetic moment have been presented: the one in terms of Poisson's 
equivalent magnetic charge, wholly analogous to the electrostatic case, 
Sec. 4.3; the other in terms of current distribution. The identity of 
these two viewpoints must now be demonstrated. 

4.6. The Magnetic Dipole. According to Sec. 4.3, a magnetized 
piece of matter has a dipole moment defined by (15), page 229, and at 
sufficient distances its scalar potential is given by (18) of the same section. 
Our first task is to determine the equivalent vector potential; i.e., a 
vector function whose curl leads to the same field as the gradient of the 
scalar <>*. We shall again let (x', y', z') be the fixed point of observation 
and locate the dipole at (x, y, z). Then 

(31) H(x', y', z') = -v'** = 1 v' [m V 
and, consequently, 

(32) v'xA_ 

To expand the right-hand side of (32), note that m is a constant vector 
or at most a function of x, y, z, whereas V' applies only to the coordinates 
x', y', z'. Furthermore 

(33) V' X V (~\ = 0, V' V = 0. 
It may be verified now without difficulty that 

(34) v' [m V (i)] - -V X [m X V Q] ; 
hence, 

(35) V< X A = - V X [m X V Q], 

(36) A + m X V = V'f(x>, y', z'), 



where / is an undetermined scalar function. The divergence of the left 
side vanishes, and consequently / may be any solution of Laplace's 



SEC. 4.7] 



MAGNETIC SHELLS 



237 



equation. However, / contributes nothing to the field H and we are at 
liberty to place it equal to zero. The vector potential of a magnetic 
dipole is, therefore, 



(37) 



A(af, if, = - 



X V'(i) = 



Equation (37) is identical with the vector potential attributed to a 
current dipole moment in the preceding paragraph. At a sufficient 
distance the field of any source magnetic matter or stationary current 
reduces to that of a magnetic dipole. This amounts to no more than a 
restatement of the fact that a magnetized region is fully equivalent to a 
current distribution of density J' = V X M, and gives mathematical 
support to Amp&re's interpretation of magnetism in terms of infinitesimal 
circulating currents. 

4.7. Magnetic Shells. We have shown thus far that the potential of 
any closed linear current can be expanded into a series of multipole 
potentials. By allowing the circuit con- 
tour to shrink to infinitesimal dimensions 
about some point (x, y, z), the potential 
can be represented everywhere but at the 
singularity (x, y, z) by the dipole term 
alone. The magnetic moment of the 
resultant elementary linear current is 



(38) 



dm = In da, 




FIG. 42. Resolution of the 
current / about the contour C 
into elementary currents. 



where n is the positive normal to the 
infinitesimal plane area da. The ele- 
mentary moment depends on the current 
and the area enclosed by the circuit, not 
upon its particular form. 

Any linear current of arbitrary size 
and configuration can be resolved 

into a system of elementary currents. Let 5, Fig. 42, be any regular 
surface spanning the circuit contour C and draw a network of intersecting 
lines terminating on C. Imagine now that about the contour of each 
elementary area there is a current / equal in magnitude to the current 
in C and circulating in the same sense. The magnetic field due to this 
network of currents is everywhere identical with that of the current / in 
the simple contour (7; for the currents along the common boundaries of 
adjacent surface elements cancel one another except on the outer contour. 
As the fineness of the division is increased, the field of each mesh current 
approaches more closely that of a dipole whose axis is oriented in the 
direction of the positive normal Clearly in the limit the field of the 



238 THE MAONETOSTATIC FIELD [CHAP. IV 

network, and consequently that of the current / about C, is identical 
with that of a dipole distribution, or double layer, over the surface S. 
The density, or moment per unit area, of the equivalent surface distribu- 
tion is constant and equal to 

(39) T = = 7n. 

In effect every linear current acts then as a magnetized shell. The 
scalar potential of such a shell has been determined in Sec. 3.16 in con- 
nection with double layers of charge. 

(40) **(*', y', z') = -1 

where ft is the solid angle from (a/, ?/', z') subtended by S. Moreover the 
potential of a double layer has been shown to be multivalued. The 
discontinuity suffered by <* as one traverses S is by (17), page 190, 

(41) </>* - <f>* = r = I] 

consequently, the line integral of H about any closed path piercing S a 
single time is 

(42) < H ds = - 

The surface S bearing the double layer or magnetic shell is completely 
arbitrary and evidently equivalent mathematically to the barrier or 
"cut" by means of which the field was reduced in Sec. 4.2 to a simply 
Connected region. 

A DIGRESSION ON UNITS AND DIMENSIONS 

4.8. Fundamental Systems. In Sec. 1.8 a dimensional analysis of 
electromagnetic quantities was based directly on Maxwell's equations 
and the adoption of the m.k.s. units was advocated on the grounds that 
they constitute a system which is both "absolute" and practical. His- 
torically the development of the various systems followed another course. 
As long as electrostatic and magnetostatic phenomena were considered 
to be wholly independent of one another, it was natural that two inde- 
pendent, absolute systems should exist for the measurement of electric 
and magnetic quantities. Discovery of the law of induction by Faraday 
and proof much later that current is no more than charge in motion estab- 
lished a connection between the two groups of phenomena and imposed 
certain conditions on the choice of otherwise arbitrary constants. 

The units and dimensions of electrostatic quantities have been com- 
monly based on Coulomb's law: the force between two charges is pro- 



SEC. 4.8] FUNDAMENTAL SYSTEMS OF UNITS 239 

portional to the product of their magnitudes and inversely proportional 
to the square of the distance between them. 

(1) F- 

Since the presence of matter does not affect the dimensions of charge, 
we assume henceforth that all forces are measured in free space. In 
" rationalized" units the proportionality constant fci is chosen equal to 
1/47T ; in "unrationalized" systems the factor 4?r is dropped. The 
constant is arbitrary and not necessarily dimensionless, so that the 
dimensions of charge can be expressed only in terms of e . From (1) 
we obtain 

(2) [q] = ecMffcir- 1 ; 

from the equation of continuity it is evident that the dimensions of 
current are those of charge divided by time. 



(3) [7] = 

From these it is a simple matter to deduce the dimensions of all other 
purely electrical quantities. 

The properties of magnetic matter can be described more naturally 
in terms of magnetic moment m than by the introduction of fictitious 
"magnetic charges." The torque exerted on a magnet or current loop 
of moment m 2 in the field B produced by a source of moment mi is 

(4) T = m 2 X B, 

and the vector B is expressed in terms of the source mi by (31), page 236. 
From these two relations one may readily verify that the dimensions 
of magnetic moment are 



(5) [m] = /I 

Equation (5) is a relation deduced directly from an expression for the 
mutual torques exerted by two magnetic dipoles, the proper magnetic 
counterpart of Coulomb's law. But this is not the only experimental 
law involving magnetic moment. It is a matter of observation that the 
moment m of a small loop or coil carrying a current 7 is proportional to 
the current and to the area of the loop. The constant of proportionality, 
which we shall call 1/y, is, like e and /z , arbitrary in size and dimensions 
so that m is subject also to the dimensional equation 

(6) [m] = y~ l [l\L* = i~ l *JM*UT-\ 

The relation of magnetic moment to current and of current to electric 
charge thus imposes a condition upon the otherwise arbitrary constants 



240 THE MAGNETOSTATIC FIELD [CHAP. IV 

o, MO and y. Whatever the choice of these three factors, their dimensions 
must be such that (5) and (6) are consistent. Upon equating (5) and 
(6), it follows that 

(7) 

The magnitude of the characteristic velocity c must be determined by 
experiment. In Chap. I it appeared as the velocity of propagation of 
fields and potentials in free space; now it occurs as a ratio of electric and 
magnetic units. To determine c without recourse to a direct measure- 
ment of velocity, a quantity is chosen that can be calculated in one system 
of units and measured in another. The capacity of a condenser, for 
example, can be calculated if its geometrical form is sufficiently simple, 
and the electrostatic capacity of any other condenser may be measured in 
terms of this primary standard by means of a capacity bridge. On the 
other hand a ballistic galvanometer measures the charge on the con- 
denser in terms of the magnetic effects produced when the condenser is 
discharged through the galvanometer. The ratio of the two values of 
capacity can be shown to be 

(8) 



where C e is the calculated electrostatic value and C m the measured value 
in magnetic units. The accepted value of c is approximately 3 X 10 8 
meters/sec. 

The various dimensional systems which at one time or another have 
been employed in the literature on electromagnetic theory are obtained 
by ascribing arbitrary values to any two of the three constants e , MO and 
7oJ the third is then determined by (7). We shall review the most 
important of these systems briefly. 

1. The Electrostatic System. If o = 1, r = 1, all quantities, both 
electric and magnetic, are expressed in electrostatic units. For MO one 
obtains from (7) the value 

(9) MO = i X 10- 20 (sec./cm.) 2 . 

The c.g.s. units in this system are based on a definition of charge from 
Coulomb's law: two unit charges of like sign concentrated at points 1 cm. 
distant repel one another with a force of 1 dyne. 

2. The electromagnetic system is obtained by putting 

(10) MO = 1, 7 = 1, 6 = i X 10- 20 (sec./cm.) 2 . 
The ratio of charge measured in the two systems is 






SEC. 4.9] COULOMB'S LAW FOR MAGNETIC MATTER 241 

where q e is the magnitude of a charge measured in electrostatic units, 
q m the magnitude of the same charge measured in e.m.u. 

3. The m.k.s. system, described in detail in Sec. 1.8, is essentially an 
electromagnetic system in which the units of mass and length have been 
adjusted such that the units of electric arid magnetic quantities conform 
to practical standards. In the form adopted in the present volume, the 
coulomb is introduced as an independent unit, leading to 

(12^ MO = 47T X 10~ 7 henry /meter, 7 = 1, 

eo = 8.854 X 10~ 12 farad/meter. 

An alternative form of the m.k.s. system puts 

(13) Mo = 47r X 10- 7 , 7 = 1, o = 8.854 X 10~ 12 (sec. /meter) 2 , 

which is dimensionally identical with the electromagnetic system. 

4. The Gaussian system has until recently been most commonly 
employed in scientific literature. It is a mixture of the electrostatic 
and electromagnetic systems obtained by putting 

(14) = 1, Mo = 1, 7 = c. 

The factors , MO drop out of all equations and electrical quantities are 
expressed in electrostatic units, magnetic quantities in magnetic units. 
The advantages of all this are rather dubious. It is fully as difficult to 
recall the proper position of the factor c entering into electromagnetic 
equations written in Gaussian units as it is to retain the factors e and /z 
from the outset. The m.k.s. system is quite as much an "absolute" or 
"scientific " system as the Gaussian, and there are few to whom the terms 
of the c.g.s. system occur more readily than volts, ohms, and amperes. 
All in all it would seem that the sooner the old favorite is discarded, the 
sooner will an end be made to a wholly unnecessary source of confusion. 

Any equation appearing in this book can be transformed to Gaussian 
units by replacing and /x each by 1/c; J and p by J/c and p/c 
respectively. 

4.9. Coulomb's Law for Magnetic Matter. In the older literature 
the magnetic system of units was commonly based on a law of force 
between magnetic "poles." The "charge" or pole strength q* was 
defined such that the product of q* and the length of a dipole was equal to 
its moment m. The dimensions of #* based on (5) are then 



(15) [<Z*] = M 

while for the law of force corresponding to Coulomb's law we obtain 

(16) '- 



242 THE MAGNETOSTATIC FIELD [CHAP. IV 

This expression differs from the one commonly assumed in that /z 
appears in the numerator rather than the denominator. 1 At the root of 
this puzzling result is the fact that the force on an element of current, 
as deduced from Maxwell's equations, Sec. 2.4, is J X B, not J X H. 
Consequently the torque on a dipole is m X B arid the force on a magnetic 
"charge" is F = j*B. 

To make (16) conform strictly to the electrostatic model it has been 
customary to write the factor ju in the denominator and to define the 
vector H by the equation F = g*H. This procedure, however, leads to 
dimensional inconsistencies in the field equations unless juo is assumed to 
be a numeric and there is little to be said in its favor. The parallel 
nature of the vectors E and B and the vectors D and H has been recog- 
nized by most recent writers. 

MAGNETIC POLARIZATION 

4.10. Equivalent Current Distributions. The equivalence of a mag- 
netized body and a distribution of current has been noted previously 
on various occasions. Let us see just how this distribution in a body of 
volume V bounded by a surface >S is to be determined. 

We shall suppose that the intensity of magnetization, or polarization, 
M includes also the permanent or residual magnetization M if any is 
present. At any point outside the body the vector potential is by (37), 
page 237, equal to 



(1) A(*', y', z') = M(z, y, z) X V dv. 
By virtue of the vector identity 

(2) Mx 



and the fundamental formula for the transformation of the volume 
integral of the curl of a vector to the surface integral of its tangential 
component, 



we see that the right-hand side of (1) may be resolved into two parts. 

fA\ A / / / A Mo I ^ X M , juo I M X n , 

(4) A(*', y', z'} = dv + -- da. 



1 SOMMERFELD, "Ueber die elektromagnetischen Einheiten," p. 161, Zeeman, Ver- 
handelingen, Martinus Nijhoff, The Hague, 1935. 



SBC. 4.11] FIELD OF MAGNETIZED RODS AND SPHERES 243 

The vector potential due to a magnetized body is exactly the same aa 
would be produced by volume and surface currents whose densities are 

(5) J' = V X M, K' = M X n. 

The validity of this result at interior as well as exterior points is 
easily demonstrated. For the surface integral certainly converges at 
all points not on /S, and the volume integral can be resolved into three 
scalar components, each of which is equivalent to the scalar potential 
of a charge distribution. 

We recall that the magnetic polarization was introduced initially, 
Sec. 1.6, as the difference of the vectors B and H. 

(6) M = - B - H. 

Mo 

At a surface of discontinuity between two magnetic media the field 
vectors satisfy "the boundary conditions of Sec. 1.13; namely, 

(7) n (B, - Bi) =0, n X (H 2 - Hi) = K, 

in which the surface current density is zero whenever the conductivity 
is finite. Then by (6) these conditions are equivalent to 

(8) n (B 2 - BO =0, n X (B, - BO = Mo (K + K'), 
where 

(9) K' = (Mi - M a ) X n. 

4.11. Field of Magnetized Rods and Spheres. A steel rod or cylinder 
of circular cross section uniformly magnetized in the axial direction will 
serve as an example. The length of the cylinder is Z, its radius a, and 
the magnetization is M . Since M is assumed constant, the volume 
density of equivalent current J' is zero. On the cylindrical wall the 
current density is K' = M X n, so that the magnetized rod is equivalent 
to a solenoidal coil of the same dimensions. The vector potential and 
field at any internal or external point can now be calculated from the 
series expansion derived in Sec. 4.5. The value of B along the axis has 
already been computed in Sec. 4.4. In particular, we find at the center 

(10) B = 



For an actual magnet the validity of our assumption of uniform 
magnetization is dubious, particularly in the vicinity of the edges. If, 
however, the length of the cylinder is either very large or very small with 



244 THE MAONETOSTATIC FIELD [CHAP. IV 

respect to the radius, the approximation is usually a good one. Thus, if 
I a so that the rod is needle-shaped, the field at the center reduces to 

(11) B = MoM . 
In case I <C a, it degenerates to a disk, with 

(12) B = 0. 

If the cylinder is soft iron instead of steel in an external field H 0) 
the permanent polarization M must be replaced by the induced mag- 
netization M and (10) is to be interpreted as the induced field BI. At the 
center of a long needle in a parallel field H > the resultant fields are 

(13) B = Bo + BI = Bo + /xoM; H = - B - M = H ; 

Mo 

the field H in the interior of the needle is the same as that which existed 
initially. In the case of the disk 

(14) B = Bo, H = Ho - M. 

The assumption of a uniform magnetization of a soft-iron cylinder 
with squarely cut ends is obviously a direct violation of the boundary 
conditions on the vectors B and H. On the other hand the induced 
magnetization of a sphere or ellipsoid in a parallel external field must be 
uniform, for the problem is wholly analogous to the electrostatic case 
considered in Sees. 3.24 and 3.27. The induced or secondary field Bt 
can now be attributed to a surface current circulating in zones concentric 
with the axis of magnetization, but the density will evidently vary with the 
latitude. If 6 is the angle made by the outward normal with the direction 
of M, 

(15) K' - |M X n] = M sin 9. 

By the Biot-Savart law the component of 67 in the direction of M is 



(16) dBT = sin* Ode = sin 3 d6. 

A 4 

When integrated this leads to 

(17) Bl 



o o 

for the induced field at the center, while for the total field one obtains 
(18) B- = B + BI = Mo(H + f M), H- - Ho - *M. 

An important corollary to this result is the value of the field at the 
center of a spherical cavity cut out of a rigidly and uniformly magnetized 



Sue. 4.12] SURFACE DISTRIBUTIONS OF CURRENT 245 

medium. The surface current distribution is the same but, since the 
normal is now directed into the sphere, the circulation is in the opposite 
direction. The induced field at the center is |/n M , and the resultant 
field is 

(19) B- = Bo - f MoM = /io(H + |M ), H- = Ho + *M , 

where Ho and B are the initial fields within the medium. Obviously 
these fields do not satisfy the customary boundary conditions on the 
normal and tangential components, and they are possible only when the 
cavity is excised after the rigid magnetization of the material. 

DISCONTINUITIES OF THE VECTORS A AND B 

4.12. Surface Distributions of Current. It follows directly from the 
investigation of Sec. 3.14 that the vectors A and B are everywhere con- 
tinuous functions of position provided the current is distributed in vol- 
ume with a density which is bounded and piecewise continuous. When, 
however, the conductivity of a body is assumed infinite, or the magnetized 
state of the body is to be represented in terms of an equivalent current 
distribution, it becomes necessary to take account of surface currents 
across which the transition of the vector B is discontinuous. Let us 
imagine a current with a finite volume density J confined to a lamellar 
region of thickness t. Suppose now that the thickness is allowed to 
shrink, but that throughout the process the total current traversing its 
cross section is maintained at a constant value. As t > 0, the volume 
density J necessarily approaches infinity. The surface current density is 
defined as the limit of a product: 



(1) K = limj, as *-0, J->oo. 

Because of the infinite value of J, this distribution constitutes a surface 
singularity. 

We shall assume that K is a bounded, piecewise continuous function 
of position on a regular surface S. The vector potential at any point 
(z', y', z') not on S is 

(2) A(,' )y ', 2 ')= 



in which r is, as usual, drawn from the element Kfc, y, z) da to the fixed 
point (x', y' y z'). Each rectangular component of A is an integral of the 
form representing the electrostatic potential of a surface distribution of 
charge. In Sec. 3.15 it was shown that such integrals are bounded, con- 
tinuous functions of the coordinates (z', y', t') at all points in space, 
including those lying on S. Both the normal and tangential components of 
the vector potential pass continuously through a single surface current layer. 



246 THE MAGNETOSTATIC FIELD [CHAP. IV 

Turning our attention now to the vector B, we have according to 
Eq. (11), page 231, at a point not on S 



(3) E(x', y', z') = K(z, y, z) x V da. 

Upon resolving K into its rectangular components, 

K = iK 9 + 
it appears that (3) is equivalent to 

(4) B(*', y', O = [i X J g X. 

+ JX 

Now an integral of the type 

(5) E(z', T/', *') = ^ J cofe y, 2) V Q da 

may be interpreted, apart from a dimensional factor , as the field due to 
a charge of density w spread over the surface S; and, as was shown in 
Sec. 3.15, it exhibits a discontinuity as the point (x' y y f , z f ) traverses the 
surface equal to 

(6) E^ - E_ = con, 

where w is the local value of the surface density at the point of transition. 
Upon applying this result to (4) it follows immediately that 

(7) B^ - B_ = /ioK X n. 

The normal component of B passes continuously through a surface layer of 
current. 

(8) n (B+ - B-) = 0. 

The tangential components of B experience a discontinuous jump defined by 

(9) n X (B+ - B_) - juon X (K X n). 

The right-hand side, by a well-known identity, transforms to 

(10) n X (K X n) = (n n)K - (n K)n. 

Since, however, K lies 'n the surface and is hence orthogonal to the unit 
normal n, (9) reduces to 

(11) n X (B+ - B_) - MO K, 

The significance of this analysis is made clearer by comparison with 
the boundary conditions deduced directly from Maxwell's equations in 



SBC. 4.13] SURFACE DISTRIBUTIONS OF MAGNETIC MOMENT 247 

Sec. 1.13. According to Sec. 4.10 a magnetized medium is equivalent to 
a volume distribution of current of density J' = V X M, but discon- 
tinuities in the magnetization must be accounted for by surface layers of 
current whose density is K' = n X (M+ M_). The discontinuity in 
the field across such a surface is 

(12) n X (B+ - B_) = Mo [K + n X (M + - M-)], 

or, since B M = H, 

Mo 

(13) n X (H+ - H_) = K. 

The continuity of the normal components of B across the boundary is 
ensured by (8). 

4.13. Surface Distributions of Magnetic Moment. The equivalence 
of a linear circuit and a surface bearing a magnetic dipole distribution was 
demonstrated in Sec. 4.7. We shall give some further attention to the 
discontinuities exhibited by the potential and field in the neighborhood of 
such singular surfaces without restriction to the case of normal magnetiza- 
tion. The vector M shall now represent the surface density of magnetiza- 
tion, the magnetic moment of the surface S per unit area. The direction 
of M with respect to the positive normal n is arbitrary. 

At any point not on S the vector potential of the distribution is by 
(37), page 237, equal to 



(14) A(x', 2/', *') = s M (x, y, z) X V da. 

We need but note the similarity of form which this expression bears to 
Eq. (3) to conclude that the vector potential experiences a discontinuity 
on passing through a surface bearing a distribution of magnetic moment, 
the amount of the discontinuity being 



(15) k - A_ = juoM X n. 
The normal component of A is continuous, 

(16) n (A+ - A_) = 0, 

but the tangential component is continuous only in case the magnetization i& 
normal to the surface. Otherwise 

(17) n X (A+ - A_) = MoM - Mo(n M)n. 



To investigate the discontinuity exhibited by the magnetic vector B 
as it passes through a surface layer of magnetic moment it will be advan- 
tageous to employ an expression of the form of Eq. (31), page 236, which 
we shall now write as 



248 THE MAGNETOSTATIC FIELD [CHAP. IV 



(18) B(x', a', z') - -. V M(*, y, *) V i da. 

** ^/s r 

The surface polarization M may be resolved into a normal and a tan- 
gential component by means of the identity 

(19) M = (n M)n + (n x M) x n, 
so that in place of (18) one has 

(20) B(*', ', z') = -g V' f (n M)n - V * da 

-~-V f f[(nx M) xn].y-da. 
^ 7r /s f 

On comparison with (16), page 189, it is clear that the first integral may 
be interpreted as the scalar potential due to a double layer of moment 
T = (M n)n, while the negative gradient of this potential at the point 
(a/, ?/', z') gives the field intensity. Now it was shown in Sec. 3.16 that 
the normal components of the field intensity approach the same limit on 
either side of such a distribution, but that the tangential components 
cross it discontinuously as expressed by Eq. (21), page 191, the amount of 
the jump in the present case being 

(21) . B t+ -B t -= - Mo ^(M.n). 

Thus the component of B along any direction tangent to a surface bearing 
a distribution of magnetic moment experiences a discontinuity on travers- 
ing the surface which is proportional to the rate of change of the normal 
component of the polarization in that same direction. 

Turning our attention next to the contribution of the second integral 
in (20), we reduce it to a more familiar form by certain transformations of 
the integrand. The identity 



(22) [(n X M) x n] V = (n X M) . n X V 

can easily be verified. Next let us resolve (n X M) into scalar com- 
ponents, so that n X M = i/i + j/ 2 + k/ 3 . We have then 

(23) [(nx M) Xn].T = i 



SBC. 4.13] SURFACE DISTRIBUTIONS OF MAGNETIC MOMENT 249 

Now if f(x, y, z) be any function which is continuous and has continuous 
first and second derivatives over a regular surface S and on its contour 
C, then 

(24) fn X Vfda = f /ds. 

Js Jc 

It follows, therefore, that if the tangential components of M satisfy the 
prescribed conditions over /S, the three surface integrals whose integrands 

are of the type i f n X V J can be transformed to integrals along any 
closed contour on S, so that 

(25) 



The remaining three terms of the integrand may be reduced by the 
relation 



= S.(l x T/. + j x T/ . + k _n-T X (0 X M) 



The second term on the right-hand side of (20) has, therefore, been 
resolved into a contour integral plus a surface integral, or 

(27) -1 V f [(n X M) X n] V - da = -i V f 
v ' 4?r Js LV ' J r 47r Jc 



c r 

/^ , f n V X (n X M) 
4x Js r 

Now the contour integral is continuous and continuously differenti- 
able at points on S which are interior to the contour. On the other 
hand, the last term of (27) might be interpreted as the field intensity at 
a point (x', y'> z') due to a surface charge distribution of density 

-n V X (n X M). 

Consequently, one must conclude that the tangential component of 
surface magnetization leads to a discontinuity in the normal components 
of B at the surface S of amount 

(28) B+ - B n - = -//on V X (n X M). 



250 THE MAGNETOSTATIC FIELD [CHAP. IV 

Differentiation is restricted to directions tangent to the surface. Con- 
sequently the vector V(M n) lies in the surface, while the vector 
V X (n X M) is normal to it. The discontinuities exhibited by the vector 
B in its transition through a surface distribution of magnetic moment can 
be expressed by the single formula: 

(29) B + - B_ = ~juo[V(n M) + V X (n X M)]. 

INTEGRATION OF THE EQUATION V X V X A = /J 
4.14. Vector Analogue of Green's Theorem. The classical treatment 
of the vector potential is based on a resolution into rectangular com- 
ponents. On the assumption that V A = 0, each component can be 
shown to satisfy Poisson/s equation and the methods developed for the 
analysis of the electrostatic potential are applicable. 

The possibility of integrating the equation V X V X A = /xj directly 
by means of a set of vector identities wholly analogous to those of Green 
for scalar functions appears to have been overlooked. Let V be a closed 
region of space bounded by a regular surface S, and let P and Q be two 
vector functions of position which together with their first and second 
derivatives are continuous throughout V and on the surface S. Then, 
if the divergence theorem be applied to the vector P X V X Q, we have 

(1) V (P X V X Q) dv = (P X V X Q) n da. 



Upon expanding the integrand of the volume integral one obtains the 
vector analogue of Green's first identity, page 165, 



(2) y (V X P V X Q - P V X V X Q) dv 

= f (P X V X Q) n da. 

/o 

The analogue of Green's second identity is obtained by an interchange 
of the roles of P and Q in (2) followed by subtraction from (2). As a 
result 



(3) y (Q V X V X P - P V X V X Q) dv 

= jf(PXVXQ-QXVXP)-ncZa. 

4.15. Application to the Vector Potential. We shall assume that the 
volume density of current J(x, y, z) is a bounded but otherwise arbitrary 
function of position. The regular surface S bounding a volume V need 
not necessarily contain within it the entire source distribution, or even 
any part of it. As in Sec. 3.4 we shall choose as an arbitrary origin 
and x = z', y = y', z = z' as a fixed point within V. 



SEC. 4.15] APPLICATION TO THE VECTOR POTENTIAL 251 

Now let P represent the vector potential A subject to the conditions 
(4) V X V X A = /J, V A = 0, 

where \i is the permeability of a medium assumed homogeneous and iso- 
tropic. The choice of Q may be made in either of two ways. It will 
be recalled that in the scalar case the Green's function \l/ satisfied Laplace's 
equation vV = and could be interpreted as the potential at (x f , y' y z') 
due to a charge 4?re located at (x, y, z). likewise the vectorial Green's 
function Q may be chosen to represent the vector potential at (z', y', z'} 
produced by a current of density 4T/ju located at (x, y, z} and directed 
arbitrarily along a line determined by the unit vector a. 



(5) Q(x, y, z] x', y', z'} = =, r = V(x' - *) 2 + (y' - 2/) 2 + (' - *) 2 - 

However the divergence of this function is not zero and consequently 

(5) fails to satisfy the condition V X V X Q = 0. On the other hand, 
the vector potential arising from a distribution of magnetic moment has 
been shown to satisfy V X V X A = everj where and an appropriate 
Green's function for the present problem is, tferefore, 

(6) Q- 

Obviously (6) may be interpreted as the vectof potential of a magnetic 

dipole of moment a. 
M 

Either (5) or (6) may be applied to the integration of (4) but the 
necessary transformations turn out to be simpler *n the case of (5) in spite 
of the divergence trouble. We have in fact 

(7) V X Q = V (~\ X a, V X V X Q = V a V 

(8) P-V X V X Q - A 



In these transformations and those that follow it is to be kept in mind that 
a is a constant vector. The left-hand side of the identity (3) can now 
be written 



252 THE MAGNETOSTATIC FIELD [CHAP; IV 

Proceeding to the transformation of the surface integrals, we have 

(10) (PX VXQ).n=JAx[vQxa]j.n 

= a V ( i j X (A X n), 

/ \ 

(11) (Q X V X P) n = 



in which B replaces V X A. The identity (3) becomes 

(12) dv = (A.n)vda + ~ X (A X n) da 



(* 
Js r 



Now the validity of this relation has been established only for regions 
within which both P and Q are continuous and possess continuous first 
and second derivatives. Q, however, has a singularity at r = and 
consequently this point must be excluded. About the point (#', y', z') a 
small sphere of radius ri is circumscribed. The volume V is now bounded 
by the surface Si of the sphere and an outer enveloping surface S as 
indicated in Fig. 25, page 166. Since V(l/r) = r/r 2 , the surface integrals 
over Si may be written 



: X (A x n) da H | n X B da. 

The integrand of the middle term is transformed to 

(13) r X (A X n) = (r - n)A - (A - n)r + A X (r X n), 

and since on the sphere r n = 1, r X n = 0, the surface integrals 
over Si reduce to 



\ f Ada + ~ f 

*1 jSi 7*1 JSi 



If A and n X B denote the mean values of the vectors A and n X B over 
the surface of the sphere, these integrals have the value 



which in the limit as n > reduces to 4rA(z', y', z 1 ). Upon introducing 
this result into (12) and transposing, we find the value of the vector 
potential at any fixed point expressed in terms of a volume integral 



SEC. 4.15] APPLICATION TO THE VECTOR POTENTIAL 253 

and of surface integrals over an outer boundary which is again denoted 
byS. 

U4> w .,, - 



The proof that the divergence of (14) at the point (x', t/', z') is zero 
is left to the reader. Clearly the surface integrals represent the con- 
tribution to the vector potential of all sources that are exterior to the 
surface 8. At all points within 7, the vector A (re', y f , z') defined by (14) 
is continuous and has continuous derivatives of all orders. Across the 
surface $, however, it is apparent from the form of the surface integrals 
that A and its derivatives will exhibit certain discontinuities. We shall 
show in fact that outside 8 the vector A is zero everywhere. 

The first surface integral in (14) may be interpreted as the contribu- 
tion to the vector potential of a surface current 

(15) K = -- n X B_ t 

in which the subscript of B_ emphasizes that this value of B is taken 
just inside the surface 8. Now in Sec. 4.12 it was shown that a surface 
layer of current does not affect the transition of the vector potential, but 
gives rise to a discontinuity in B of amount 

(16) n X (B+ - B_) = jiK. 

Upon replacing K by its value from (15) it is clear that just outside S 

(17) n X B + = 

In like manner the second surface integral is equivalent to the vector 
potential of a distribution of magnetic surface polarization of density 

(18) M = - A_ X n. 

The normal component of A passes continuously through such a layer, 
but the tangential component is reduced discontinuously to zero, as we 
see on substituting (18) into (17), page 247 

(19) n X (A + - A_) = -n X A- + [n . (n X A_)]n, 
whence 

(20) n X A+ = 0, 



254 THE MAONETOSTATIC FIELD [CHAP. IV 

The mathematical significance of the last term of (14) is apparent, 
but it is difficult to imagine a physical distribution of current or magnetic 
moment of the type called for. This is the form of integral which we 
have associated with the field intensity of a surface charge of density 
(A_-n), and which leads to a discontinuity in the normal component 
specified by 

(21) n - (A+ - A..) = n A., 
as a consequence of which we conclude that 

(22) n A+ = 0. 

Thus far we have demonstrated that on the positive side of the closed 
surface S the tangential and normal components of A and the tangential 
component of B are everywhere zero. It follows at once, however, that 
the normal component of B must also vanish over the positive side of S; 
for the normal component of the curl A involves only partial derivatives in 
directions tangential to the surface. Furthermore, we need but apply 
(14) itself to the region external to S to prove that A, and consequently 
B, must vanish everywhere. Current and magnetic polarization are 
absent outside F, since their effect is represented by the surface integrals. 
Then, since n X B+, n X A+ and n A+ are all zero, it follows from (14) 
that A(z', y' t z') must vanish at all points outside S. 

When Q = v(l/r) X a is chosen in place of (5) as a Green's function, 
it can be shown without great difficulty that 

(23) B(*', y>, O = J^ J X V dv - J^ (n X B) X V (j) da 



This is the extension of the Biot-Savart law to a region of finite extent 
bounded by a surface S. The contribution of currents or magnetic 
matter outside 8 to the field within is accounted for by the two surface 
integrals. 

BOUNDARY-VALUE PROBLEMS 

4.16. Formulation of the Magnetostatic Problem. A homogeneous, 
isotropic body is introduced into the constant field of a fixed and specified 
system of currents or permanent magnets. Our problem is to determine 
the resultant field both inside and outside the body. In case the current 
density at all points within the body is zero, the secondary field arising 
from the induced magnetization can be represented everywhere by a 
single-valued scalar potential <* and the methods developed for the 



SBC. 4.16] FORMULATION OF THE MAGNETOSTATIC PROBLEM 255 

treatment of electrostatic problems apply in full. A schedule for the 
solution may be drawn up as follows. 

The scalar potential of the primary field is <ftf . In case the primary 
source is a current distribution <$ is multivalued but this in no way 
affects the determination of the induced field </>?,. The resultant potential 
is </>* = (* + 0*. The permeability of the body will be denoted by jm 
and that of the homogeneous medium in which it is embedded by ju 2 . 
Then a function <#>* must be constructed such that: 

(1) V 2 <? = 0, at all points not on the boundary; 

(2) </>f is finite and continuous everywhere including the boundary; 

(3) Across the boundary the normal derivatives of the resultant potential 
<* satisfy the condition 



the subscripts + and implying that the derivative is calculated 
outside or inside the boundary surface respectively. The induced 
potential <* is, therefore, subject to the condition 



in which f is a known function of position on the boundary satisfying 
the condition 



(4) 

(5) At infinity <f>f must vanish at least as 1/r 2 , so that r*4>? remains 
finite as r > <*> y for there is no free magnetic charge and conse- 
quently <f must vanish as the potential of a dipole or multipole of 
higher order. 

In case the body carries a current, the interior field cannot be repre- 
sented by a scalar potential and the boundary-value problem must be 
Solved in terms of a vector potential. Such a case arises, for example, 
when an iron wire carrying a current is introduced into an external mag- 
netic field. The distribution of the current in the stationary state is 
unaffected by the magnetic field. Its determination is in fact an electro- 
static problem. The vector potential of the primary sources is A while 
the potential of the induced and permanent magnetization of the body 
and of the current which it may carry will be denoted by AI. This 
function AI is subject to the following conditions: 

(1) V X V X AI = jwj, at points inside the body where the current 
density is J ; 



256 THE MAONETOSTATIC FIELD [CHAP. IV 

(2) V X V X Ai =* 0, at all other points not on the boundary S; 

(3) V AI = 0, at all points not on S; 

(4) Ai is finite and continuous everywhere and passes continuously 
through the boundary surface (Sees. 4.10 and 4.12); 

(5) Across the boundary the normal derivatives of the potential Ai as 
well as those of the total potential A satisfy 



/dA 

V *r 



= * o( + ~ } x ' 

in w/iic/t n is 2/ie outward normal and where M_ is Me polarization 
of the body and M+ that of the medium just outside the boundary. 

Since M+ and ML are determined at least in part by the field itself this 
relation is usually of no assistance in the determination of AI. In its 
place the customary boundary condition on the tangential components 
of the total field must be applied. 



0, which imposes a rela 



(6) n X (H+ - H _) = n X ( B+ - ^ B_ ) 

\ Mi / 

tion between the derivatives of A in a specified coordinate system; 

(7) As r > GO the product rAi remains finite. 

4.17. Uniqueness of Solution. The proof that there is only one func- 
tion <t>* satisfying the conditions scheduled above was presented in 
Sec. 3.20. A corresponding uniqueness theorem for the vector potential 
may be deduced from the identity (2) of page 250. Let us put 
p = Q == A and assume first that within V\ bounded by S the current 
density is zero. Then V X V X A = and 

CO f (V X A) 2 <fo = - f A (n X V X A) da. 

JVl 4/0 

From the essentially positive character of the integrand on the left it 
follows that if A is zero over the surface S, then B = V X A is zero 
everywhere within the volume V\. Hence A is either constant or at 
most equal to the gradient of some scalar ^. But since A is zero on S, 
the normal derivative d^/dn is also zero over this surface and it was 
shown in Sec. 3.20 that this condition entails a constant value of ^ 
throughout V\. Consequently, if A vanishes over a closed surface, it 
vanishes also at every point of the interior volume. It is also clear that 
the vector function A is uniquely determined in Vi by its values on S. 
For if there existed two vectors Ai and A2 which assumed the specified 
values over the boundary, their difference must vanish not only over 8 
but also throughout V\. 

The condition V X A = 0, A = V^, throughout Vi can be estab- 
lished also by the vanishing of V X A, or of the tangential vector 



9ac. 4.18] FIELD OF A UNIFORMLY MAGNETIZED ELLIPSOID 257 

n X V X A over S. In this ease, however, it does not necessarily follow 
that A is everywhere zero. If the two functions V X AI = Bi and 
V X A 2 = B 2 are identical on S, then BI and B 2 are identical at all 
interior points and AI and A 2 can differ at most by the gradient of a 
scalar function. 

In case there are currents present within V\ the vector potential A 
is resolved into a part A' due to these currents and a part A" due to 
external sources. The vector A' is uniquely determined by the current 
distribution, while the values of A" or its curl over S determine A" at 
all interior points. 

The vector potential is regular at infinity and consequently the proof 
applies directly to the region F 2 exterior to S. The vector B is uniquely 
determined within any domain by the values of its tangential component 
n X B over the boundary. 

PROBLEM OF THE ELLIPSOID 

4.18. Field of a Uniformly Magnetized Ellipsoid. An ellipsoid whose 
semiprincipal axes are a, b } c is uniformly and permanently magnetized. 
The direction of magnetization is arbitrary, but since the magnetization 
vector can be resolved into three components parallel to the principal 
axes we need consider only the case in which M is constant and parallel 
to the a-axis. 

In view of the uniformity of magnetization p* = V M = at 
all points inside the ellipsoid. The potential <* of the magnet is due to a 
"surface charge" of density w* = n M . The external medium is 
assumed in this case to be empty space. The problem is now fully 
equivalent to that of the polarized dielectric ellipsoid treated in Sec. 
3.27. From Eqs. (27), (32), and (42) the potential 4>i due to the polariza- 
tion P x may be found in terms of P x and the parameters of the ellipsoid. 
Upon dropping the factor e and replacing P x by M Qx one obtains 

(i) K-AJI* A 



as the magnetic scalar potential at points inside the ellipsoid; at all 
external points 



AW A* 

(2) tf = 

The field inside the ellipsoid is 



H * = ~ = - 



258 THE MAGNETOSTATIC FIELD [CHAP. IV 

Consequently if the magnetization is parallel to a principal axis, the field 
H inside the ellipsoid is constant, parallel to the same axis, but opposed 
to the direction of magnetization. 

4.19. Magnetic Ellipsoid in a Parallel Field. An ellipsoid of homo- 
geneous, isotropic material is introduced into a fixed and uniform parallel 
field Ho. The ellipsoid is free of residual magnetism and we shall suppose 
that for sufficiently weak fields its permeability jui is constant, as is also 
the permeability ^2 of the external medium. 

The determination of the field can again be adopted directly from the 
electrostatic case, for upon reference to Sec. 4.16 it will be noted that 
the boundary conditions imposed upon <* are identical with those to be 
satisfied by <t> when ju is substituted for . The resultant magnetic 
potential at an interior point of the ellipsoid is, therefore, 



, " uz~ H- Qy ?/ 

9 ~" ~T~ ~1~ ~ 



1 + o 0*1 "~ P2)Al 1 + 



1 I aC f 

1 + 9 (Ml 

2^2 

outside the ellipsoid <* is obtained from (38), page 213, by replacing 
by M and adding the contributions induced by Hoy and H Oz . 

The torque exerted by the field on the ellipsoid may likewise be 
determined directly from (54), page 216, after a similar substitution. 
The importance of these results is fundamental. There are experimental 
methods of measuring small torques with great precision. The magnetic 
field near the center of a long solenoid is very nearly parallel and uniform, 
and its intensity in terms of the current can be calculated. Thus the 
permeability or susceptibility of the test sample can be determined with 
great accuracy. 

CYLINDER IN A PARALLEL FIELD 

4.20. Calculation of the Field. As an example of the application of 
the vector potential, let us consider the following simple problem. A 
cylinder or wire of circular cross section and permeability m is embedded 
in a medium of permeability ^2- The wire is infinitely long, has a radius 
a, and carries a current I. The external field B is directed transverse 
to the axis of the wire and is everywhere parallel and uniform. We 
wish to calculate first the resultant field at points inside and outside 
the cylinder. 

The x-axis of the reference system will be chosen parallel to the 
vector BO, while the z-axis is made to coincide with the axis of the cylinder 



SEC. 4.20] CALCULATION OF THE FIELD 259 

and the direction of the current, Fig. 43. A vector potential is to be 
found which satisfies the conditions 

(1) V X V X A = 0, (r> a), 

(2) V X V X A = n J, (r < a). 

The current density has only a z-component. 

(3) J, = J V = 0, J, = ^ (r< a). 

The vector potential may be resolved into two parts: the potential A 
of the applied field, and the secondary potential AI due in part to the 




Fio. 43. Cylinder in a uniform magnetostatic field. 



current I, in part to the induced magnetization. Now clearly B can 
be derived from a vector potential directed along the z-axis. If i 3 is 
the unit vector in the positive z-direction, then 

(4) Ao = isB Q y = i 3 jB r sin 9. 

Moreover AI is also oriented along this same axis, for the magnetization 
can be represented by equivalent volume and surface currents. Cer- 
tainly the induced magnetization lies in the transverse plane and con- 
sequently the equivalent currents J' = V X MI and 

K' = n X (M 2 - MO 

of Sec. 4.10 are directed along the cylinder. Since the vector potential 
is always parallel to the current, AI X = Ai v = 0. The expansion of (2) 
is now a simple matter. From (85), page 50, we have 



in which the subscript z has been dropped. 

A general solution must first be found for the homogeneous equation 
obtained by putting the right-hand member of (5) equal to zero. The 
separation of such an equation was discussed in Sec. 3.21, and since the 
vector potential is a single-valued function of we write down at once 



260 THE MAONETOSTATIC FIELD [CHAP. IV 



(6) A' = (a n cos n6 + b n sin n0)r n + ^ (c n cos n0 + d n ,cos n0)r~ n 

n*0 n=0 

Moreover the vector potential must be finite everywhere in space and 
consequently inside the cylinder the coefficients c n and d n are zero. 

The general solution of (5) is obtained by the usual method of adding 
to (6) any particular solution of (5). Since r 2 J depends only on r, we 
shall assume that this particular solution A" is independent of 9. 



> '" - 

Consequently at all interior points of the cylinder the resultant vector 
potential is 



(9) A- = - ( J + ( a * cos n * + 6 sin ne > n - 

V \ / n ~Q 

Since the vector potential of the induced magnetization is regular at 
infinity, outside the wire we put a n = b n in (6). To this must be 
added the vector potential of the current I, which in Eq. (20), page 233, 
was shown to be /* 2 //27r In 1/r, and lastly the contribution of the external 
field itself from (4). The resultant vector potential at any point outside 
the cylinder is, therefore, 

(10) A+ = B Q r sin 9 + ~ In ~ + ^ ( c cos n + d sin ne)r~ n . 

n = 

Next, the coefficients of the series expansions are determined from the 
condition that the vector potential and tangential components of H shall 
be continuous across the boundary r = a. From the expansion of the 
curl in cylindrical coordinates, page 51, we have 

/ in p _ 1 dA z R SA Z 

(11) B '~r~de' 5 < = ~~dF' 

so that the boundary conditions imposed on the vector potential are 

(12) A--A+, 

N ' ' z 

The result of equating coefficients of like terms from (9) and (10) is then 

a = 0, Co = (2ji2 In a MI) -*-> 

7, - 2Ml P /7 _ Mi M2 
01 = - : - JDQ, aj = - : - 

Ml + M2 Ml + M2 



Sac. 4.21J FORCE EXERTED ON THE CYLINDER 261 

and all other coefficients zero, 



By means of (11) the field may be calculated. 



cos 
2 



sn 0- 



The nature of the field inside the cylinder is made somewhat clearer by 
a transformation to rectangular coordinates. 

(17) H x = H r cos - ffo sin 0, H = H r sin + He cos 0. 

rr- 2JSo / _ 7 



The magnetization of the cylinder is given by the relation 



Mo 
Mo 



X. 



Mo 

Evidently the magnetization induced by the applied field B is in the 
direction of the positive x-axis. Upon this is superposed the magnetiza- 
tion induced by the current L 

4.21. Force Exerted on the Cylinder. If the medium supporting the 
wire is fluid, or a nonmagnetic solid, the force per unit length exerted by 
the field can be computed from (15), page 155. Resolved into rec- 
tangular components, this becomes 

F* = I M 2 (#*#r - J ff 2 cos B] r d6. 
(20) Jo \ 2 / 

f 2 - / i \ 

F v = I M2 ( H y H r -H* sin 6 } r dO. 

The rectangular components of the field outside the cylinder are first 
calculated from (15) and (17). 



262 THE MAGNETOSTATIC FIELD (CHAP. IV 

H+ = ff + Ml 7^ Ho cos 20 - ^ sin 0, . 
(21) 1.1 + M.r* arr 

H+ = Ml , M 'g. Ho sin 20 + H- cos 0, 

* Ml ' a 



while H 2 is equal to the sum of the squares of these two components. 






47rV 2 



Upon introducing (21) and (22) into (20) and evaluating the integrals, 
one obtains the components of the total force exerted by the field upon 
the wire and the current it carries. 

(23) F x = 0, F v = M 2#o/. 

That the induced magnetic moment of the wire contributes nothing to 
the resultant force follows from the uniformity of the applied field. If 
this field, on the other hand, were generated by the current in a neighbor- 
ing conductor, the distribution of induced magnetization would no longer 
be symmetrical; a force would be exerted on the cylinder even in the 
absence of a current /. 

Problems 

1. An infinitely long, straight conductor is bounded externally by a circular cylin- 
der of radius a and internally by a circular cylinder of radius b. The distance between 
centers is c, with a > b + c. The internal cylinder is hollow. The conductor carries 
a steady current 7 uniformly distributed over the cross section. Show that the field 
within the cylindrical hole is 



H 



- 6 2 ) 



and is directed transverse to the diameter joining the two centers. 

2. Two straight, parallel wires of infinite length carry a direct current 7 in opposite 
directions. The conductivity a- of the wires is finite. The radius of each wire is a and 
the distance between centers is 6. 

a. Using a bipolar coordinate system find expressions for the electrostatic poten- 
tial and the transverse and longitudinal components of electric field intensity at 
points inside and outside the conductors. 

b. Find expressions for the corresponding components of magnetic field intensity. 

c. Discuss the flow of energy in the field. 

3. A circular loop of wire of radius a carrying a steady current 7 lies in the st/-plane 
with its center at the origin. A point in space is located by the cylindrical coordinates 



PROBLEMS 263 

r, <f>, z, where x = r cos <, ?/ r sin <. Show that the vector potential at any point 
in the field is 



= OM/ f 

2r J 



(a 2 + r 2 + z 2 - 2ar cos , 



where /b 2 = 4ar/[(a -f r) 2 + z 2 ], and K and E are complete elliptic integrals of the 
first and second kinds. 

Show that when (r 2 -f z 2 )* a, this reduces to Eq. (37), p. 237, for the vector 
potential of a magnetic dipole. 

4. From the expression for the vector potential of a circular loop in Problem 3 
show that the components of field intensity are 

Hr = - . - , , 5 + r* + z* 



2w r[(a + 



H - 

' 



2*- [(a 



a-r- 8 ] 

+ (-^+^ J" 



6. The field of two coaxial Hclmholtz coils of radius a and separated by a distance 
a between centers is approximately uniform in a region near the axis and halfway 
between them. Assume that each coil has n turns, that the cross section of a coil is 
small relative to a, and that each coil carries the same current. From the results of 
Problem 4 write down expressions for the longitudinal and radial components of H at 
points on the axis and at any point in a plane normal to the axis and located midway 
between the coils. What is the field intensity at the mid-point on the axis? Find 
an expansion for the longitudinal component H z in powers of z/a valid on the axis in 
the neighborhood of the mid-point, and corresponding expressions for U t and H t in 
powers of r/a valid over the transverse plane through the mid-point. 

6. Two linear circuits Ci and C 2 carry steady currents /i and 7 2 respectively. 
Show that the magnetic energy of the system is 



1 fc f c ! 

/ Ci j Ca 



where ds\ and dsz are vector elements of length along the contours and 7*12 the distance 
between these elements. The permeability ju is constant. 

The coefficient of mutual inductance is defined by the relation 



- -- 
4?r J c\ 



7. Show that the mutual inductance of two circular, coaxial loops hi a medium of 
constant permeability is 



Li, - 



- | E\ 



264 THE MAGNETOSTATIC FIELD [CHAP. IV 

where a and 6 are the radii of the loops, c the distance between centers, K and E the 
complete elliptic integrals of the first and second kinds, and k is defined by 

4a6 



(a + 6) 2 + c 2 

When the separation c is very small compared to the radii and a 6 show that this 
formula reduces to 



^Ma(ln "J ~ 



where d - [(a - &) + c 2 ]*. 

8. The coefficient of self-inductance L\\ of a circuit carrying a steady current I\ is 
defined by the relation 



where T\ is the magnetic energy of the circuit. Hence L\\ can be calculated from 
either of the expressions 



' ll " T\ J J ' A dv ~ 7? J 



provided B is a linear function of H. The volume integral in the first case is extended 
over the region occupied by current, in the second over the entire field. That portion 
of LU associated with the energy of the field inside the conductor is called the internal 
self-inductance Z/ n . 

Show that the internal self-inductance of a long, straight conductor of constant 
permeability /*i is 

L' n = henry s /meter. 

Sir 

9. Show that the self-inductance of a circular loop of wire of radius R and cross- 
sectional radius r is 



where /ui is the permeability of the wire and /z 2 that of the external medium, both being 
assumed constant. 

10. A toroidal coil is wound uniformly with a single layer of n turns on a surface 
generated by the revolution of a circle r meters in radius about an axis R meters from 
the center of the circle. Show that the self-inductance of the coil is 

henrys. 

11. A circular loop of wire is placed with its plane parallel to the plane face of a 
semi-infinite medium of constant permeability. Find the increase in the self -induc- 
tance of the loop due to the presence of the magnetic material. 

Show that if the plane of the loop coincides with the surface of the magnetic 
material, the self-inductance is increased by a factor which is independent of the form 
of the loop. 

12. Find an expression for the change in self-inductance of a circular loop of wire 
due to the presence of a second circular loop coaxial with the first. 



PROBLEMS 



265 



13. An infinitely long, hollow cylinder of constant permeability MI is placed in a 
fixed and uniform magnetic field whose direction is perpendicular to the generators of 
the cylinder. The potential of this initial field is <* H^r cos 0, where r and 6 
are cylindrical coordinates in a transverse plane with the axis of the cylinder as an 
origin. The cross section of the cylinder is bounded by concentric circles of outer 
radius a and inner radius 6. The permeability of the external and internal medium has 
the constant value M2. Show that the potential at any interior point is 



1 - 



/MI + M,V _ /&Y 

\Mi - M2/ \a/ 



HQ r cos 8, 



(r < 6). 



14. A hollow sphere of outer radius a and inner radius 6 and of constant permeabill 
ity MI is placed in a fixed and uniform magnetic field Ho. The external and internal 
medium has a constant permeability M2. Show that the internal field is uniform and is 
given by 



H 



1-1- 



1 - 



(MI 



20*1 ~ M2) 2 



bV 

a 



Ho, 



(r < 6). 



Discuss the relative effectiveness of a hollow sphere and a long hollow cylinder for 
magnetic shielding. 

16. A magnetic field such as is observed on the earth's surface might be generated 
either by currents or magnetic matter within the earth, or by circulating currents 
above its surface. Actual measurement indicates that the field of external sources 
amounts to not more than a few per cent of the total field. Show how the contribu- 
tions of external sources may be distinguished from those of internal sources by meas- 
urements of the horizontal and vertical components of magnetic field intensity on the 
earth's surface. 

16. Assume that the earth's magnetic field is due to stationary currents or mag- 
netic matter within the earth. The scalar potential at external points can then be 
represented as an expansion in spherical harmonics. Show that the coefficients of 
the first four harmonics can be determined by measurement of the components of 
magnetic field intensity at eight points on the surface. 

17. A winding of fine wire is to be placed on a spheroidal surface so that the field 
of the coil will be identical with that resulting from a uniform magnetization of the 
spheroid in the direction of the major axis. How shall the winding be distributed? 

18. A copper sphere of radius a carries a uniform charge distribution on its sur- 
face. The sphere is rotated about a diameter with constant angular velocity. Cal- 
culate the vector potential and magnetic field at points outside and inside the sphere. 

19. A solid, uncharged, conducting sphere is rotated with constant angular velocity 
w in a uniform magnetic field B, the axis of rotation coinciding with the direction of 
the field. Find the volume and surface densities of charge and the electrostatic 
potential at points both inside and outside the sphere. Assume that the magnetic 
field of the rotating charges can be neglected. 

20. The polarisation P of a stationary, isotropic dielectric in an electrostatic field 
E is expressed by the formula 

P - ( - o)B. 



266 THE MAONETOSTATIC FIELD [CHAP. IV 

If each element of the dielectric is displaced with a velocity v in a magnetic field the 
polarization is 

P - ( - eo )(E+v XB), 

at least to terms of the first order in 1/c = \/ OMO. 

A dielectric cylinder of radius a rotates about its axis with a constant angular 
velocity w in a uniform magnetostatic field. The field is parallel to the axis of the 
cylinder. Find the polarization of the cylinder, the bound charge density appearing 
on the surface, and the electrostatic potential at points both inside and outside the 
cylinder. 

21. A dielectric sphere of radius a in a uniform magnetostatic field is rotated with 
a constant angular velocity co about an axis parallel to the direction of the field. Using 
the expression for polarization given in Problem 20, find the densities of bound volume 
and surface charge and the potential at points inside and outside the sphere. 

22. Show that the force between two linear current elements is 



4?r 



where r is a unit vector directed along the line r joining <isi and d$ 2 from d$i toward 

G?S 2 . 

23. A long, straight wire carrying a steady current is embedded in a semi-infinite 
mass of soft iron of permeability /t at a distance d from the plane face. The wire is 
separated from the iron by an insulating layer of negligible thickness. Find expres- 
sions for the field at points inside and outside the iron. 

24. A long, straight wire carrying a steady current is located in the air gap between 
two plane parallel walls of infinitely permeable iron. The wire is parallel to the walls 
and is assumed to be of infinitesimal cross section. Find the field intensity at any 
point in the air gap. Plot the force exerted on the conductor per unit length for a cur- 
rent of one ampere as a function of its distance from one wall. 

// * * * '///// ^* A- l n &> straight conductor of infinitesimal 
Iron "///< cross section lies in an infinitely deep, parallel-sided 
slot in a mass of infinitely permeable iron. The thick- 



p^ i ness of the slot is I and the conductor is located by the 
parameters s and h as in the figure. Calculate the 

//A \ f ,,,, *\ f , , ^ e ^ intensity at points within the slot and the force 

^K/^^ on unit Ien 8 th of conductor per ampere. (Cf. Hague, 

" Electromagnetic Problems in Electrical Engineering," 
and Linder, /. Am. lust. Elec. Engrs., 46, 614, 1927.) 

26. An infinite elliptic cylinder of soft iron defined by the equation 



is placed in a fixed and uniform magnetic field whose direction is perpendicular to the 
generators of the cylinder. Find expressions for the magnetic scalar potential at 
internal and external points and calculate the force and torque exerted on the cylinder 
per unit length. Discuss the case of unit eccentricity in which the cylinder reduces 
to a thin slab. 

27. A particle of mass m and charge q is projected into the field of a magnetic 
dipole. 



PROBLEMS 267 

a. Write the differential equations for the motion of the particle in a system of 
spherical coordinates whose origin coincides with the center of the dipole. 

6. Show that the component of the angular momentum vector in the direction of 
the dipole axis is constant. 

c. Discuss the trajectory in the case of a particle initially projected in the equa- 
torial plane. 

28. A small bar magnet is located near the plane face of a very large mass of soft 
iron whose permeability is M. The magnet is located by its distance from the plane 
and the angle made by its axis with the perpendicular passing through its center. 
Considering only the dipole moment of the magnet, find the force and torque exerted 
on it by the induced magnetization in the iron. 

29. Two small bar magnets whose dipole moments are respectively mi and ma are 
placed on a perfectly smooth table. The distance between their centers is large 
relative to the length of the magnets. At any instant the dipole axes make angles 0i 
and 02 with the line joining their centers. 

a. Calculate the force exerted by mi on ma and the torque on ni2 about a vertical 

axis through its center. 
6. Calculate the force exerted by m 2 on mi and the torque on mi about a vertical 

axis through its center. 
c. Calculate the total angular momentum of the system about a fixed point in the 

plane. Is it constant? 

30. Show that the force between two small bar magnets varies as the inverse 
fourth power of the distance between centers, whatever their orientation in space. 

31. A magnetostatic field is produced by a distribution of magnetized matter. 
There are no currents at any point. Show that the integral 



U 



B H dv = 



if extended over the entire field. 

32. If J(a, y, z) is the current density at any point in a region V bounded by a closed 
surface S, show that the magnetic field at any interior point x r , y f , z' is 



where r \/Oe' #) 2 -f- (y f y) 2 H- (z r z) 2 , and that at exterior points the value 
of the surface integral is zero. 

33. Determine the field of a magnetic quadrupole from the expansion of the vector 
potential given by Eq. (23), p. 234. Give a geometrical interpretation of the quad- 
rupole moment in terms of infinitesimal linear currents. 



CHAPTER V 
PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA 

Every solution of Maxwell's equations which is finite, continuous, 
and single-valued at all points of a homogeneous, isotropic domain 
represents a possible electromagnetic field. Apart from the stationary 
fields investigated in the preceding chapters, the simplest solutions 
of the field equations are those that depend upon the time and a single 
space coordinate, and the factors characterizing the propagation of 
these elementary one-dimensional fields determine also in large part 
the propagation of the complex fields met with in practical problems. 
We shall study the properties of plane waves in unbounded, isotropic 
media without troubling ourselves for the moment as to the exact nature 
of the charge and current distribution that would be necessary to establish 
them. 

PROPAGATION OF PLANE WAVES 

6.1. Equations of a One-dimensional Field. It will be assumed for 
the present that the medium is homogeneous as well as isotropic, and 
of unlimited extent. We shall suppose, furthermore, that the relations 

(1) D = E, B = /*H, J = <rE, 

are linear so that the medium can be characterized electromagnetically 
by the three constants , /*, and a. If the conductivity is other than zero 
any initial free-charge distribution in the medium must vanish spontane- 
ously (Sec. 1.7). In the following, p will be put equal to zero in dielectrics 
as well as conductors. The Maxwell equations satisfied by the field 
vectors are then 

-JTT 

(I) V X E + ^ = 0, (III) V - H = 0, 

*\Tf 

(II) VXH-e^~-(7E = 0, (IV) V - E = 0. 

We now look for solutions of this system which depend upon the time 
and upon distance measured along a single axis in space. This preferred 
direction need not coincide with a coordinate axis of the reference system. 
Let us suppose, therefore, that the field is a function of a coordinate 
f measured along a line whose direction is defined by the unit vector n. 
The rectangular components n X) Uy, n of this unit vector are obviously 

268 



SEC. 5.1] 



EQUATIONS OF A ONE-DIMENSIONAL FIELD 



269 



the direction cosines of the new coordinate axis J". Our assumption 
implies that at each instant the vectors E and H are constant in direction 
and magnitude over planes normal to n. These planes are defined by 
the equation 

(2) r n = constant, 

where r is the radius vector drawn from the origin to any point in the 
plane as indicated by Fig. 44. 




FIG. 44. Homogeneous plane waves are propagated in a direction fixed by the unit vector n. 

Since the fields are of the form 
(3) E - E(T, 0, H - H(r, 0, 

the partial derivatives with respect to a set of rectangular coordinates 
may be expressed by 

d d d d e d 



From these we construct the operator v and so obtain simplified expres- 
sions for the curl and divergence. 



(6) 
(7) 



V X E = n X 



&E 



(a x 



In virtue of these relations, the field equations assume the form 
(I) n X |? + /* ^ = 0, (III) n ^? = 0, 

OC ut u\ 

C/Xl UjCj v* ^ x-r-r-rv C/JC/ 

i X ^r - -TT - o-E = 0, (IV) n = 0. 
o ol o$ 



270 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

This system is now solved simultaneously by differentiating first (I) 
with respect to f and then multiplying vectorially by n. (II) is next 
differentiated with respect to t. We have 



(8) 



ff at ' 



upon elimination of the terms in H, the vector E is found to satisfy the 
equation 



A similar elimination of E leads to an identical equation for H. 

, 1A , 3 2 H 3 2 H 3H . 

(10) -^--^-^--^^-=0. 

Since n ( n X ^7 1 is identically zero, it follows from (I) that 



c 

_, =o. Taken together with (III), this gives 
at 



We are forced to conclude that a variation of the f-component of H with 
respect to either f or t is incompatible with the assumption of a field which 
is constant over planes normal to the f-axis at each instant. Equation 
(11) does admit the possibility of a static field in the f -direction but, since 
at present we are concerned only with variable fields, we shall put Hf = 0. 
Likewise, it follows from (II) that 



(12) n 

this together with (IV) leads to 






(13) n -c + -E<tt + d =n.dE + Edt =0. 

\dt e ar / \ / 

The component of E normal to the family of planes, therefore, satisfies 
the condition 

(14) 

which upon integration gives 
(15) 



SBC. 5.1] EQUATIONS OF A ONE-DIMENSIONAL FIELD 271 

where E^ is the longitudinal component of E at t = and T = /<r is the 
relaxation time defined on page 15. If the conductivity is finite, the 
longitudinal component of E vanishes exponentially: a static electric 
field cannot be maintained in the interior of a conductor. According to 
(15) there may be a component E{ in a perfect dielectric medium, but 
this particular integral does not contain the time. 

Equations (11) and (13) prove the transversality of the field. The 
vectors E and H of every electromagnetic field subject to the condition (3) lie 
in planes normal to the axis of the coordinate f . 

Let us introduce a second system of rectangular coordinates , 77, f , 
whose origin coincides with that of the fixed system x, y, z and whose 
f-axis is oriented in the direction specified by n. With respect to this 
new system the vector E has the components E^ Jg,, and E f = 0. Both 
EZ and EH satisfy (9), which can be solved by separation of the variables. 
Let 



Then 



where k 2 is the separation constant. The general solution of the 
equation in /i is 

(18) /!() = Ae*t + Ber**, 

where A and B are complex constants; for/ 2 we shall take the particular 
solution 

(19) M) = Or*. 
Then p must satisfy the determinantal equation 

(20) p' - ? p + *! = 0. 

There exists a fixed relation between p and the separation constant fc 2 ; 
the value of either may be specified, whereupon the other is determined. 
From (18) and (19) may be constructed a particular solution of the 
form 



(21) ES = 

likewise, for the other rectangular component, 

(22) E, = Ei^-^ + 



272 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 



The four complex constants EU . . . E%, are the components of two 
complex vector amplitudes lying in the 77 plane. Combining (21) and 
(22), one obtains 



(23) 



E = 



-v + 



This vector solution of (9) in turn is introduced into the field equations 
to obtain the associated magnetic vector H. Since H must have the 






n 



PIG. 45. Relative directions of the electric and magnetic vectors in positive and negative 

waves. 

same functional dependence on f and , we can write 

(24) H = Hie**-* + Htf-**-*, 

and then determine the constants Hi, H 2 in terms of Ei and E 2 . 



(25) = i 



dt 



= -pH. 



These derivatives are introduced into (I) and lead to 

(26) (ifcn X Ei - pMHi)e^-"< - (ifcn X E 2 + pvK^e-**-* = 0. 



The coefficients of the exponentials must vanish independently and, 
hence, 



(27) 



Hi = n X Ei, 



= n X 2* 



In terms of the rectangular components, 

"" ik 









SBC. 5.2] PLANE WAVES HARMONIC IN TIME 273 

Since E n X E = 0, it follows that E H = 0. The electric and magnetic 
vectors of afield of the type defined by Eq. (3) are orthogonal to the direction n 
and to each other. These mutual relations are illustrated in Fig. 45. 

5.2. Plane Waves Harmonic in Time. There are now two cases to 
be considered, dependent upon the choice of p and fc, which it will be well 
to treat separately. Let us assume first that the field is harmonic in 
time, and that therefore p is a pure imaginary. The separation constant 
is then determined by (20). 

(29) p = ico, ft 2 = /zeco 2 + ijjuru. 

In a conducting medium fc 2 , and hence k itself, is complex. The sign of 
the root will be so chosen that the imaginary part of k is always positive. 

(30) k = a + iff. 

The amplitudes EU . . . E*, are also complex and will be written now in 
the form 



where the new constants a x . . . 6 2 , #1 . . . ^2 are real. In virtue of 
these definitions we have for the ^-component of the vector E 

(32) EZ = aie-ft**-"**'* + a^t-****"'-'* . 

Since (32) is the solution of a linear equation with real coefficients, both 
its real and imaginary parts must also be solutions. The real part of (32) 
is 



(33) ES = aie-*r cos (ut - af - 0i) + a^ cos (wt + af - 2 ). 

The phase angles 0i and 2 are arbitrary and consequently a choice of the 
imaginary part of (32) does not lead to a solution independent of (33). 
In the same way the Ty-component of E is obtained from (22). 



(34) # = fctf-tt cos (at - af - ^i) + bzePt cos (w + ctf - 



The components of the associated magnetic field are found from (28). 
We now have 



ik a + i ft V 2 + 

= - - = - ! 



, i 
-s tan" 1 - 



The complex ^-component of H is 

(36) ff = Hi^r+iw-^o + H u ^r 

upon substitution of the appropriate values of His an d #2$, we obtain for 



cos (erf - af - 0i - 7) 



cos (erf + af - 2 - 7). 




274 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
the real part 

-v/^2 JL /Q2 

COS (cot Qjf tyl 7) 



if* cos (erf 
/zco 

Similarly for the rj-component, 



(38) 



These solutions correspond obviously to plane waves propagated 
along the f-axis in both positive and negative directions. Consider first 
the most elementary case in which all amplitudes except a\ are zero in a 
nonconduct ng medium. Then a and consequently ft are zero and the 
field is represented by the equations 

(39) Et = ai cos (erf af 0i), HI = ai cos (erf af 0i). 

fJ.00 

This field is periodic in both space and time. The frequency is eo/27r = v 
and the period along the time axis is 2?r/co = T. The space period is 
called the wave length and is defined by the relation 

fAf\\ ^ \ ^fl" 

(40) a ) X = 

A a 

The argument <i = erf af 0i of the periodic function is called the 
phase and the angle 0i, which will be determined by initial conditions, is 
the phase angle. At each instant the vectors E and H are constant over 
the planes f = constant. Let us now choose a plane on which the phase 
has some given value at t = 0, and inquire how this plane must be dis- 
placed along the f-axis in order that its phase shall be invariant to a 
change in t. Since on any such plane the phase is constant, we have 

(41) $1 = co af 0i = constant, d<j>i = w dt a d = 0. 

The surfaces of constant phase in a field satisfying (39) are, therefore, 
planes which displace themselves in the direction of the positive f-axis 
with a constant velocity 



v is called the phase velocity of the wave. It represents simply the 
velocity of propagation of a phase or state and does not necessarily 
coincide with the velocity with which the energy of a wave or signal is 



SEC. 5.2J PLANE WAVES HARMONIC IN TIME 275 

propagated. In fact v may exceed the critical velocity c without violating 
in any way the relativity postulate. 
In a nonconducting medium 

(43) a = co A/Ate, v = 

where c is the velocity of the wave in free space and *, K m are the electric 
and magnetic specific inductive capacities. In optics the ratio 
n = c/v ac/co is called the index of refraction. Since in all but ferro- 
magnetic materials K m is very nearly unity, the index of refraction should 
be equal to the square root of K e . This result was first established by 
Maxwell and was the basis of his prediction that light is an electro- 
magnetic phenomenon. Marked deviations from the expected values of 
n were observed by Maxwell, but these were later accounted for by the 
discovery that the inductive capacity does not necessarily maintain at 
high frequencies the value measured under static or quasi-static condi- 
tions. A functional dependence of n on the frequency results in a 
corresponding dependence of the phase velocity and leads to the phe- 
nomena known as dispersion. 

At a given frequency the wave length is determined by the properties 
of the medium. 

(44) X = - = H 

v n 

where X is the wave length at the same frequency in free space. In all 
nonionized media n > 1, and consequently the phase velocity is decreased 
and the wave length is shortened. 

Referring again to (39), we see that the relation of the magnetic 
to the electric vector is such that the cross product E X H is in the 
direction of propagation. The magnetic vector is propagated in the 
same direction with the same velocity, and in a nonconducting medium 
is exactly in phase with the electric vector. Their amplitudes differ 
by the factor 



(45) = ^|i = 2.654 X 10- 3 ^, (, = 0). 

If in Eqs. (33) to (38) all amplitudes except a 2 are put equal to zero, 
another particular solution is found which in nonconducting media 
reduces to 

(46) EI = a 2 cos (at + af - 2 ), ff f = a 2 J- cos (at + af - 0,). 

This field differs from the preceding only in its direction of propagation. 
The phase is < 2 = wt + af 2 and the surfaces of constant phase are 



276 PLANE WAVES IN UNBOUNDED, ISOTROP1C MEDIA [CHAP. V 

propagated with the velocity v = w/a in the direction of the negative 
f-axis. 

The two particular solutions whose amplitudes are bi and 6 2 represent 
a second pair of positive and negative waves, both characterized by 
electric vectors parallel to the rj-axis. 

Let us remove now the restriction to perfect dielectric media and 
examine the effect of a finite conductivity. It is apparent in the first 
place that both electric and magnetic vectors are attenuated exponen- 
tially in the direction of propagation. Waves traveling in the negative 
direction are multiplied by the factor e~w, but since f decreases in the 
direction of propagation this too represents an attenuation. Not only 
does a conductivity of the medium lead to a damping of the wave, but 
it affects also its velocity The constants a and & can be calculated in 
terms of p,, *, and <r by squaring (30) and equating real and imaginary parts 
respectively to the real and imaginary terms in (29). 



(47) cf - P 

Upon solving these relations simultaneously, we obtain 






Ambiguities of sign that arise on extracting the first root are resolved 
by noting that a and ft must be real. 

The planes of constant phase are propagated with a velocity 



-1/2 



which increases with frequency so long as the constants K e , Km, and or are 
independent of frequency. Attenuation of amplitudes in a surface of 
constant phase is determined by the attenuation factor ft which also 
increases with increasing frequency. The complex factor k will be referred 
to as the propagation constant, and its real part a may be called the phase 
constant, although this term is applied also to the angles 6 and ^. 

The effect of frequency and conductivity on the propagation of plane 
waves is most easily elucidated by consideration of two limiting cases. 
Examination of (48) and (49) shows that the behavior of the factors 
a and is essentially determined by the quantity <r 2 /c 2 co 2 . Now the 



SEC. 5.2] PLANE WAVES HARMONIC IN TIME 277 

total current density at any point in the medium is 

(51) J = <rE + ^ = (* - twe)E f 

Of 

whence it is apparent that or/we is equal to the ratio of the densities of 
conduction current to displacement current. 



2 



Case I. ~2~2 ^ ! The displacement current is very much greater 

than the conduction current. This situation may arise either in a 
medium which is but slightly conducting, or in a relatively good con- 
ductor such as sea water through which is propagated a wave of very 
high frequency. Expansion of (48) and (49) in powers of <r 2 / 2 2 gives 

(52) = 

(53) " 



One will remark that to this approximation the attenuation factor is 
independent of frequency. In a given medium the attenuation approaches 
asymptotically a maximum defined by (53) as the frequency is increased. 

2 

Case II. i?^ 1- The conduction current greatly predominates 

e cu 

over the displacement current. This is invariably the case in metals, 
where a is of the order of 10 7 mhos/meter. Not much is known about the 
inductive capacity of metals but there is no reason to believe that it 
assumes large values. Since the magnitude of e is probably of the order 
of 10~ n , the displacement current could not possibly equal the conduction 
current at frequencies less than 10 17 , lying in the domain of atomic 
phenomena to which the present obviously does not apply. For a and ft 
we obtain the approximate formula 

(54) a = ft = J^j- = 1.987 X 10~ 3 Vw^. 

An increase of frequency, permeability, or conductivity contributes in 
the same way to an increase in attenuation. The phase velocity also 
increases with frequency, but decreases with increasing a or K m . Thus 
the higher order harmonics of a complex periodic wave are constantly 
advancing with respect to those of lower order. 

The amplitudes of the electric and magnetic vectors of a plane wave 
a*e related by 

(55) IHI = 




278 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

(6) 

In poorly conducting media expansion of (56) gives 

(57) V* + P* ^ 2.654 x 10 -3 / ( i + 0.807 X 10 20 

JUCO \ AC m \ 

whereas in good conductors, 

< 58 ) ^T^ - ^ = 355 ' 8 >/ (a* 0' 

In a perfect dielectric the electric and magnetic vectors oscillate in 
phase; if the medium is conducting, the magnetic vector lags by an 
angle y. 

(59) 




If T~2^ ^ ^ s ra ^ reduces to unity; hence the magnetic vector of a 

plane wave penetrating a metal lags behind the electric vector by 45 deg. 

6.3. Plane Waves Harmonic in Space. The assumption that p in 
Eq. (20) is a pure imaginary leads to complex values of k and to electro- 
magnetic fields which are simple harmonic functions of time. A field 
which at any given point on the f-axis is a known periodic function of t 
can be resolved by a Fourier analysis into harmonic components propa- 
gated along the f-axis as just described. The time variation at any other 
fixed point can then be found by recombining the components at the 
point in question. In place of the variation of the field with t at a certain 
point f , one may be given the distribution with respect to f at a specified 
value of t and asked to determine the field at any later instant. A 
harmonic analysis must then be made with respect to the variable f . 

Let the separation constant k 2 be real. Then p is a complex quan- 
tity determined by 

/*m * , 

(60) p=! _ 



Vfc 2 <r 2 
-- T"V the electric vector defined 
/* 4e 2 

in (23) assumes the form 



(61) E - Etf-S- 8 + 



SBC. 5.4] POLARIZATION 279 

If now <7 2 /4 2 < fc 2 //i, the quantity iq is a pure imaginary and the field 
may be interpreted as a plane wave propagated along the f-axis with a 
phase velocity 

(62) . - * - f 

The amplitude of oscillation at any point f decreases exponentially with t, 
the rate of decrease being determined essentially by the relaxation time 

T = /ff. 

When <r 2 /4e 2 > fc 2 /Ate, the quantity iq is real and there is no propaga- 
tion in the sense considered heretofore. The field is periodic in f but 
decreases monotonically with the time. There is no displacement along 
the space axis of an initial wave form: the wave phenomenon has degener- 
ated into diffusion. 

5.4. Polarization. Inasmuch as the properties of the positive and 
negative waves differ only in the direction of propagation, we shall con- 
fine our attention at present to the positive wave alone. Furthermore, 
since the attenuating effect of a finite conductivity in an isotropic, 
homogeneous medium enters as an exponential factor common to all 
field components, it plays no part in the polarization and will be neglected. 
Let us suppose then that the amplitudes and phase constants of the 
rectangular components of E have been specified and investigate the 
locus of |E| = V^* 2 + EI? in the plane f = constant. 

To determine this locus one must eliminate from the equations 

(63) EI = a cos (<t> + 0), E, = b cos (</> + ^), # f = 0, 

the variable component < = af ut of the phase. To this end (63) 
is written in the form 

W V 

(64) ^ = cos (</> + 0), -^ = cos (0 + 0) cos 5 - sin (<f> + 0) sin 5, 

where 6 = ^ 0. Upon squaring, the periodic factor cos (< + 0) can 
be eliminated and it is found that the rectangular components must satisfy 
the relation 



The discriminant of this quadratic form is negative, 

faa\ ^ cos2 5 * 4 . 2 A 

(66) rrr --- ^2 = -&-* sin 2 6^0, 

a 2 o 2 a 2 o 2 a 2 o 2 

and the locus of the vector whose components are E^ U, is, therefore, 
an ellipse in the {rj-plane. 



280 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 



In this case the wave is said to be elliptically polarized. The points 
of contact made by the ellipse with the circumscribed rectangle are found 
from (65) to be ( a, 6 cos 6) and (a cos d, b). In general the 
principal axes of the ellipse fail to coincide with the coordinate axes , 17, 

but the two systems may be brought into 
coincidence by a rotation of the coordinate 
system about the f-axis through an angle & 
defined by 




(67) 



tan 2tf = 



2ab 
a 2 - I 



cos 6. 



7 



FIG. 46. Polarization ellipse. 



When the amplitudes of the rectangular 
components are equal and their phases differ 
by some odd integral multiple of 7r/2, the 
polarization ellipse degenerates into a circle. 



(68) 



= a 2 , (a = 6, 



d = 



mir 

it' 



m = 1, 3, 



The wave is now said to be circularly polarized. Two cases are recog- 
nized according to the positive or negative rotation of the electric vector 
about the f-axis. It is customary to describe as right-handed circular 
polarization a clockwise rotation of E when viewed in a direction opposite 
to, that of propagation, looking towards 
the source. 

By far the most important of the 
special cases is that in which the polari- 
zation ellipse degenerates into a straight 
line. This occurs when 5 = WTT, where 
m is any integer. The locus of E in the 
??-plane then reduces to a straight line 
making an angle & with the -axis de- 
fined by 





Right-handed Left-handed 

FIG. 47. Circular polarization, 
the circles indicating the locus of E. 
The direction of propagation is nor- 
mal to the page and toward the 
observer. 



(69) 



tantf = ? = (-!) 



- 
a 



The wave is linearly polarized. The magnetic vector of a plane wave is 
at right angles to the electric vector and H oscillates, therefore, parallel 
to a line whose slope is ( l) m a/6. It is customary to define the polariza- 
tion in terms of E and to denote the line (69) as the axis of linear polariza- 
tion. In optics, however, the orientation of the vectors is specified 
traditionally by the " plane of polarization/ 7 by which is meant the plane 
normal to E containing both H and the axis of propagation. 



SEC. 5.5] ENERGY FLOW 281 

5.5. Energy Flow. The rate at which energy traverses a surface in an 
electromagnetic field is measured by the Poynting vector, S = E X H 

ik 
defined in Sec. 2.19. Since by (27) H = n X E, the sign depending 

upon the direction of propagation, it is apparent that the energy flow is 
normal to the planes of constant phase and in the direction of propagation. 
To calculate the instantaneous value of the flow one must operate 
with the real parts of the complex wave functions. The mean value of 
the flow can be quickly determined, on the other hand, by constructing 
the complex Poynting vector S* = ^E X H as in Sec. 2.20. For sim- 
plicity let us consider a plane wave linearly polarized along the x-axis 
and propagated in the direction of positive z. Then 

E x = e-**+i<*-H-) 

& \A* 2 + # 2 * - f 
H = a ~ e~^ z ~~ l(az ' 



The complex flow vector is, therefore, 

1 ~ A/rv 2 -4- ft 2 

* v ~ 



whose real part represents the energy crossing on the average unit area 
of the 7/-plane per second. 

fl 2 A Ay 2 I 2 

(72) & = ~ V ^ P e-w* cos 7. 

v ' 2 /KO 

The factor cos 7 arising from the relative phase displacement between 
E and H may be expressed in terms of a and /3. 



(73) tan 7 = -> cos 7 = 

/v 

f*7A\ Q /? %pz n 2 

(74) > T> a - 

^jjUCO 

According to Poynting's theorem the divergence of the mean flow vector 
measures the energy transformed per unit volume per second into heat- 
In the present instance 

(75) v . fc _ - _ a e ^ > 

which in turn is obviously equal to the conductivity times the mean 
square value of the electric field vector E. 



282 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

6.6. Impedance. In a world which outwardly is all variety, it is 
comforting to discover occasional unity and tempting to speculate upon 
its significance. To the untutored mind a vibrating set of weights 
suspended from a network of springs appears to have little in common 
with the currents oscillating in a system of coils and condensers. But 
the electrical circuit may be so designed that its behavior, and the vibra- 
tions of the mechanical system, can be formulated by the same set of 
differential equations. Between the two there is a one-to-one corre- 
spondence. Current replaces velocity and voltage replaces force; and 
mass and the elastic property of the spring are represented by inductance 
and the capacity of a condenser. It would seem that the " absolute 
reality," if one dare think of such a thing, is an inertia property, of which 
mass and inductance are only representations or names. 

Whatever the philosophic significance of mechanical, electrical, and 
chemical equivalences may be, the physicist has made good use of them 
to facilitate his own investigations. The technique developed in the last 
thirty years for the analysis of electrical circuits has been applied with 
success to mechanical systems which not long ago appeared too difficult 
to handle, and mechanical problems of the most complicated nature are 
represented by electrical analogues which can be investigated with 
ease in the laboratory. Not only the methods but the concepts of 
electrical circuits have been extended to other branches of physics. 
Certainly the most important among these is the concept of impedance 
relating voltage and current in both amplitude and phase. This idea 
has been applied in mechanics to express the ratio of force to velocity, and 
in hydrodynamics, notably in acoustics, to measure the ratio of pressure 
to flow. 

The extension of the impedance concept to electromagnetic fields is 
not altogether new, but it has recently been revived and developed in a 
very interesting paper by Schelkunoff. 1 The impedance offered by a 
given medium to a wave of given type is closely related to the energy 
flow, but to bring out its complex nature we may best start with an 
analogy from a one-dimensional transmission line, as does Schelkunoff. 

Let z measure length along an electric transmission line and let 
V = Voe"', I = lQe~ iut be respectively the voltage across and the cur- 
rent in the line at any point z. The quantities Vo, 7o are functions 
of z alone. The resistance of the line per unit length is R and its indue* 
tance per unit length is L. There is a leakage across the line at each 
point represented by the conductance G and a shunt capacity C. The 
series impedance Z and the shunt admittance Y are, therefore, 



(76) Z = R - uoL, Y = G - icoC, 

1 SCHELKUNOFF, Bell System Tech. /., 17, 17, January, 1938. 



SEC. 5 61 IMPEDANCE 283 

while the voltage and current are found to satisfy the relations 1 

< ^=- ZI > Tz=~ YV - 

These equations are satisfied by two independent solutions which repre- 
sent waves traveling respectively in the positive and negative directions. 

, 7jn 1 1 = A i tt *-*", 7i = Zo/i, 

U * ; la = 

where 

(79) k 

k is the propagation constant and Z the characteristic impedance of the 
line. 2 

Consider now a plane electromagnetic wave propagated in a direction 
specified by a* unit vector n. Distance in this direction will again be 
measured by the coordinate f and we shall suppose that the time enters 
only through the factor e~ iut . Whereas voltage and current are scalar 
quantities, E and H are of course vectors. To establish a fixed con- 
vention for determining the algebraic sign, the field equations will be 
written in such a way as to connect the vector E with the vector H X n 
which is parallel to E and directed in the same sense. 

(80) = ico M (H X n), ~ (H X n) - i(w + 6r)E. 
of df 

By analogy with (77) 

(81) Z = -iw/i, Y = -*( + id). 
The propagation constant is 

(82) k = i 



as in (29), page 273, while the intrinsic impedance of the medium for 
plane waves is defined by Schelkunoff as the quantity 

(83) Z a -- 




1 See for example, Guillemin, "Communication Networks/' Vol. II, Chap. II, 
Wiley, 1935. 

2 These results differ in the algebraic sign of the imaginary component from those 
usually found in the literature on circuit theory due to our choice of e~^ 1 rather than 
c* w< . Obviously this choice is wholly arbitrary. Although it is advantageous to 
use e* wt in circuit theory, we shall be concerned in wave theory primarily with the space 
rather than the time factor. The expansions in curvilinear coordinates to be carried 
out in the next chapter justify the choice of e~ IM *. 



284 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
In free space this impedance reduces to 

(84) Z<> = A p = 376.6 ohms. 



Assuming n to be in the direction of propagation so that the distinction 
between positive and negative waves is unnecessary, the relation between 
electric vectors becomes 

(85) n X^E = Z H, E = Z H X n. 

There is an intimate connection between the intrinsic impedance 
and the complex Poynting vector. 

(86) S* = ^E X H = ^E x ^-^ = -|- (E . E)n, 

* ^ Zo 2Zo 

and, consequently, 



GENERAL SOLUTIONS OF THE ONE-DIMENSIONAL WAVE EQUATION 

In the course of this chapter we have investigated certain particular 
solutions of the field equations which depend on one space variable and 
the time. Owing to the linear character of the equations these particu- 
lar solutions may be multiplied by arbitrary constants and summed to 
form a general solution, about which we now inquire. In virtue of the 
infinite set of constants at our disposal it must be possible to construct 
solutions that satisfy certain prescribed initial conditions. We shall 
show, for example, that the distribution of a field vector as a function of f 
may be specified at a stated instant t = o, and that the field is thereby 
uniquely determined at all subsequent times; or the variation of a field 
vector as a function of time may be prescribed over a single plane f = f o 
and the field thus determined at all other points of space and time. 
The means at our disposal for the investigation of general integrals of a 
partial differential equation of the second order fall principally into two 
classes: the methods of Fourier and Cauchy, and those of Riemann and 
Volterra. The latter have come to constitute in recent years the most 
fundamental approach to the theory of partial differential equations, and 
their application to the theory of wave propagation has been developed 
in a series of brilliant researches by Hadamard. 1 But although the 
method of characteristics, first proposed by Riemann, affords a deeper 
insight into the nature of the problem, it proves less well adapted as yet 

1 HADAMARD, "Legons sur la propagation des ondes," A. Hermann, Paris, 1903; 
"Lectures on Cauchy's Problem," Yale University Press, 1923. 



SEC. 5.7] ELEMENTS OF FOURIER ANALYSIS 285 

to the requirements of a practical solution than the methods of harmonic 
analysis which will occupy our attention in the present section. 

6.7. Elements of Fourier Analysis. For the convenience of the 
reader, who presumably is to some degree conversant with the subject, 
we shall set forth here without proof the more elementary facts concerning 
Fourier series and Fourier integrals. 

Consider the trigonometric series 

(1) -rjr + (i cos x + bi sin x) + (a n cos nx + b n sin nx) + , 

with known coefficients a n and b n , and let it be assumed that the series 
converges uniformly in the region ^ x < 2ir. Then (1) converges 
uniformly for all values of x and represents a periodic function /(x) with 
period 2?r. 

(2) ^ f(x + 2 ? r) = /(x). 

The coefficients of the series may now be expressed in terms of /(x). 
In virtue of the assumed uniformity of convergence, (1) may be multi- 
plied by either cos nx or sin nx and integrated term by term. In the 
domain < x < 2-rr, the trigonometric functions are orthogonal; that 
is to say, 



(3) 



p2ir /2r 

I cos nx cos mx dx = 0, sin nx sin mx dx = 0, 

Jo Jo 



r 

Jo 



cos nx sin mx dx = 0, (m ^ n] 

/o 

Since 

/*2ir /*27T 

(4) I cos 2 nx dx = j sin 2 mx dx = ?r, 
Jo Jo 

it follows that 

i r 2 * 

a n = ~ I /(x) cos nx dx, 
v Jo 

(5) (m = 0, 1, 2 -.. ), 



1 f 27r 

= ~ I /(x) si 
* Jo 



sin nx dx. 



Inversely let us suppose that a function /(x) is given. Then in a 
purely formal manner one may associate with f(x) a " Fourier series" 
defined by 

00 

(6) f(x) ~ - + (a n cos nx + b n sin nx), 



286 PLANE WAVES IN UNBOUNDED, ISOTROPlC MEDIA [CHAP. V 

the coefficients to be determined from (5). The right-hand side of (6). 
however, will converge to a representation of /(x) in the domain 
^ x < 2ir only when the function /(x) is subjected to certain con- 
ditions. The whole question of the convergence of Fourier series is an 
e, and even the continuity of /(x) is not sufficient to 



ensure it. It turns out, fortunately, that under such circumstances 
the series can nevertheless be summed to represent accurately the func- 
tion within the specified interval. 1 A statement of the least stringent 
conditions to be imposed on an otherwise arbitrary function in order 
that it may be expanded in a convergent or summable trigonometric 
series would necessitate a lengthy and difficult exposition; since we are 
concerned solely with functions occurring in physical investigations, wo 
can content ourselves with certain sufficient (though not necessary) 
requirements. We shall ask only that in the interval < x < 2ir the 
function and its first derivative shall be piecewise continuous. A function 
/(x) is said to be piecewise continuous in a given interval if it is con- 
tinuous throughout that interval except at a finite number of points. If 
such a point be x , the function approaches the finite value /(x + 0) as 
x is approached from the right, and /(x 0) as it is approached from 
the left. At the discontinuity itself the value of the function is taken to 

k ,t .,, ,. /(xo + 0) + /(x - 0) 

be the arithmetic mean - Z___^L^ - '. 

z 

The Fourier expansion is particularly well adapted to the representa- 
tion of functions which cannot be expressed in a closed analytic form, 
but which are constituted of sections or pieces of analytic curves not 
necessarily joined at the ends. In view of the admissible discontinuities 
in the function or its derivative, the fact that two Fourier series represent 
the same function throughout a subinterval does not imply at all that 
they represent one and the same function outside that subinterval. 
It is thus essential that one distinguish between the representation of a 
function and the function itself. The expansion in a trigonometric 
series indicated on the right-hand side of Eq. (6) represents and is 
equivalent to the piecewise continuous function /(x) within the region 
^ x ^ 27T. Outside this domain the values assumed by the series 
are repeated periodically in x with a period 27r, whereas the f unction /(x) 
may behave in any arbitrary manner. Only when /(x) satisfies the 
additional relation (2) will it coincide with its Fourier representation 
over the entire domain -co < x < < . 

If within a specified interval the Fourier series of /(x) is known to be 
uniformly convergent, it may be integrated term by term and the series 
thus obtained placed equal to the integral of /(x) between the same limits. 

1 WHITTAKER and WATSON, "Modern Analysis/' 4th ed., Chap. IX, Cambridge 
University Press, London. 



SEC. 5.7] ELEMENTS OF FOURIER ANALYSIS 287 

The uniform convergence of a Fourier series is in itself not sufficient to 
justify its differentiation term by term and the equating of the derived 
series to df/dx. If furthermore /(x) is discontinuous at some point 
within an interval, its Fourier series certainly is not uniformly convergent 
everywhere within that interval, and consequently the Fourier expansions 
of its integral and of its derivative demand special attention. 1 

It is usually advantageous to replace (6) by an equivalent complex 
series of exponential terms of the form 



(7) /(*) = ]g c n 

n = 

whose coefficients are determined from 



(8) Cn = ^ Jo f^ e ~ in da > (" = 0, 1, 2 - ) 
Since 

(9) e inx = cos nx + i sin nx, 

it is clear that the complex coefficients c n are related to the real coeffi- 
cients a n and b n by the equations 

2c n = a n - ib n , (n> 0), 

(10) 2c = a , (n = 0), 

2c n = a_ n + ib-n, (n < 0). 

By an appropriate change of variable the Fourier expansion (7) 
may be modified to represent a function in the region I < x < I. 



-If 

O7 I 
*/ 



-<* 

f(oi)e l da - 



Now this stretching of the basic interval or period from 2?r to 2Z suggests 
strongly the possibility of passing to the limit as the basic interval 
becomes infinite and thereby obtaining a Fourier representation of a 
nonperiodic function for all real values of x lying between oo and + oo . 
We shall assume that throughout the entire region oo < x < oo , f(x) 
and its first derivative are piecewise continuous, that at the discontinui- 
ties the value of the function is to be determined by the arithmetical 

/OO 
f(x) dx is absolutely convergent. 
- CO 

or in other words, that the integral | \f(x)\ dx exists. Let w/l = Aw. 

J w 

1 These questions are thoroughly treated by Carslaw, " Introduction to the Theory 
of Fourier's Series and Integrals," Chap. VIII, Macmillan, 1921. 



288 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP, V 
Then (11) may be written 



(12) /(*) = ^ 2j A ^ J_j /()**" A " ( ~ da. 

(* CO 

On the other hand the definite integral I <(w) du is defined as the 

J 00 

limit of a sum as Aw > 0: 

f 00 ^h 

(13) I $(w) du = lim ^ <l>(n Aw) Aw. 



As Z approaches infinity, Aw approaches zero and we may reasonably expect 
the limit of (12) to be 



(14) 



=-r (^ du ( /(a)e<-> da. 

^7T J _ oo J - oo 



This is the Fourier integral theorem according to which an arbitrary 
function, satisfying only conditions of piecewise continuity and the 

existence of \ \f(x)\ dx, may be expressed as a double integral. 1 

J 00 

It will be noted that the result has been obtained by a purely formal 
transition to the limit which makes the existence of the representation 
appear plausible but does not confirm it. A rigorous demonstration ia 
beyond the scope of these introductory remarks. 

If f(x) is real, the imaginary part of the Fourier integral must vanish 
and (14) then reduces to 

i r r* 

(15) f(x) = - I du I /(a) cos u(x a) da. 

K Jo J - 

Since 

(16) cos u(x a) = cos ux cos ua + sin ux sin wa, 

it is apparent that the Fourier integral of a real, even function f(x) = 
/(*) is 

2 r r 

(17) f(x) - I COS UX I /(a) cos wa da du, 

* Jo Jo 

and that of a real, odd f unction /( x) = /(#) is 



(18) 



2 r * r 

/(#) = - I sin wx I /(a) sin ua da du. 
* Jo Jo 



1 As a general reference on the subject the reader may consult "Introduction to 
the Theory of Fourier Integrals," by E. C. Titchmarsh, Oxford University Press, 1937. 



SEC. 5.7] ELEMENTS OF FOURIER ANALYSIS 289 

From these formulas we see that the Fourier integral of a function may 
be interpreted as a resolution into harmonic components of frequency 
w/2rr over the continuous spectrum of frequencies lying between zero 
and infinity. In (18), for example, one may consider the function 

(19) Q(U) = J- f(a) sin ua da 

\* Jo 

as the amplitude or spectral density of f(x) in the frequency interval u 
to u + du. Then 

12 r l/tt) 

(20) f(x) = */- I g(u) smuxdu. 

\TT Jo 

The relation between f(x) and g(u) 
is reciprocal. g(u) is said to be the 



Fourier transform of f(x), and f(x) is FlQ . 4 8. step function, 

likewise the transform of g(u). In 

the more general case of Eq. (14) one may write for the spectral density 
of f(x) the function 

(21) g(u) = = I f(x)e- iux dx, 
and hence for f(x) the reciprocal relation 

(22) /(*) = 



-= . 

An extensive table of Fourier transforms has been published by Campbell 
and Foster. 1 

The application of the Fourier integral may be illustrated by several 
brief examples of practical interest. Consider first the discontinuous 
step function defined by 

f(x) = 1, when \x\ < I, 

(23) /(x) = i, when \x\ = I, 

f(x} = 0, when \x\ > L 

The function is real and even, so that we may employ (17). The trans- 
form is 

/o>i\ / N /2 f l , /2sinuZ 

(24) g(u) = ^- J Q cos wx dx = ^ ^ , 

and the Fourier integral 

/^x ^/ N /2 f * / x , 2 f sin ul cos ux , 

(25) f(x) = A /~ I 0(u) cos uxdu = - I du. 

\7T JO 7T JO W 

1 CAMPBELL and FOSTER: "Fourier Integrals for Practical Applications," Bell 
Telephone System Tech. Pub., Monograph B-584, 1931. Appeared in earlier form in 
Bell System Tech. J., October, 1928, pp. 639-707. 



290 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
Let us take next for /(re) the " error function" 

(26) /(*) = e'^. 

To find its Fourier transform we must calculate the integral 



(27) g(u) = A /- I e 2 cos ux dx t 

\*r Jo 

which is clearly equal to the real part of the complex integral 

(28) g(u) = ^ 

On completing the square, (28) may be written 

(29) *<*>->! '"*/.' 



/9 - u * C Q'/ 3 ' 

= v^ e ~ 2at U e " 2 ^- 

* ^ o 



The path of integration follows the imaginary axis from iu/a 2 to 
and then runs along the real axis from to co . The imaginary part of 
the integral arises solely from the interval iu/a 2 ^ < 0, and since 
we are concerned only with the real part the lower limit may be taken 
equal to zero. The result is a definite integral whose value is well known: 

f _S! 

(30) e 2 dp = 

Jo 

so that the transform of e 2 is 

(31) g(u) = e~ 
Then/(x), which is reciprocally the transform of g(u), is 

/2 f * e~^"* - 

(32) f(x) = A /- I cos ux du = e 2 . 

\ir Jo 

In the particular case a = 1, Eq. (32) becomes a homogeneous integral 

equation satisfied by the function e 2 . The pair of functions defined 
by (31) and (32) have other properties more important for our present 
needs. Let us modify (31) slightly to form the function 



(33) fife a)-- 



SEC. 5.7] 



ELEMENTS OF FOURIER ANALYSIS 



291 



The area under this curve is equal to unity, whatever the value of the 
parameter a; 

(34) " S(x, a) dx = 1, 



as follows directly from (30). Now let a become smaller and smaller. 
The breadth of the peak grows more and more narrow, while at the same 
time its height increases in such a manner in the neighborhood of x = 
as to maintain the area constant, as indicated in Fig. 49. In the limit 
as a > 0, the curve shrinks to the line x = where it attains infinite 
amplitude. A singularity has been generated, an impulse function 
bounding unit area in the immediate neighborhood of x = 0. A unit 



S(x,a) 




-2 

FIQ. 49. The impulse function S(x, a). 

impulse function which vanishes everywhere but at the point x = x is 
represented by 



(x-xo) 



(35) 



SQ(X XQ) = li 



Any arbitrary function F(), subject to the usual conditions of continuity, 
can now be expressed as an infinite integral. 



(36) 



F(Q = 



- x)F(x) dx. 



It is apparent from (32) that the transform of Jbhe impulse function SQ(X) 
is a straight line displaced by an amount l/\/27r from the horizontal axis. 
As a final illustration of the Fourier integral theorem consider a 
harmonic wave train of finite duration. Such a pulse might result from 
closing and then reopening a switch connecting a circuit to an alternating 
current generator, or represent the emission of light from an atom in the 
course of an energy transition. Let 



(37) 



/(*) = 0, 

f(t) = COS C0 , 



when 
when 



292 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

To facilitate the integration we shall take f(t) equal to the real part of 

T 
e ot in the domain \t\ < -= and use Eqs. (21) and (22). 



J2 r s i n v wu 

,"""- -^ 

The Fourier integral of /(O corresponds now to the real part of (22), or 

/,. (coo -o>)r 



(39) /(f) = I . cos 

C0 - CO 



J f ' 
! 
L 



(co ~ co)T . (coo + co)T 
sin ~ sin p: 



co co coo -r co 



COS 



Equation (39) may be interpreted as a spectral resolution of a function 
which during a finite interval T is sinusoidal with frequency co . The 
amplitude of the disturbance in the neighborhood of any frequency co is 
determined by the function 

. (co - co)T 
sm^ s ^- 

(40) A(w) = =- 

V ' 7T C0 CO 

The amplitude vanishes at the points 

(41) co = coo - np> (n = 1, 2 ), 

in the manner indicated by Fig. 50, and has its maximum value at 
co = COQ. As the duration T of the wave train increases, the envelope 
of the amplitude function is compressed horizontally until in the limit, 
as y _ > oo y the entire disturbance is confined to the line co = co on the 
frequency spectrum. The simple harmonic variations that enter into 
so many of our discussions are mathematical ideals; the oscillations of 
natural systems, whether mechanical or electrical, are finite in duration, 
and the associated waves are periodic only in an approximate sense. 
We shall discover shortly that in the presence of a dispersive medium the 
entire character of the propagation may be governed by the duration 
of the wave train. 

5.8. General Solution of the One-dimensional Wave Equation in a 
Nondissipative Medium. To acquire some further facility in the use of 
the Fourier integral it will repay us to pause for a moment over the ele- 
mentary problem of finding a general solution to the wave equation in a 



SEC. 5.8] 



GENERAL SOLUTION 



293 



nonconducting medium. Let ^ represent either the x- or ^-component 
of any electromagnetic vector. Then \f/ satisfies 



(42) 



ay 



dt 2 



According to Sec. 5.2, a particular solution of (42) is represented by 



(43) 



Be 



The coefficients A and B are arbitrary and may depend on the frequency 
w; that is to say, we associate with each harmonic component an appro- 
priate amplitude which may be indicated by writing A(o>) and (o>). 




FIG. 60. The amplitude function A(w) = - Shl (c 



Now the general solution of (42) is obtained by summing the particulai 
solutions over a range of co. In case ^ is to be a periodic function, ths 
sum will extend over a discrete set of frequencies. In general the wave 
function is aperiodic in both space and time, and the frequency spectrum 
is consequently continuous. 



(44) 



*(,<)= f" 

J c 



+ B()e 



Let us suppose that over the plane 2 = the values of the function ^ 
and of its derivative in the direction of propagation are prescribed func- 
tions of time. 



<45) 



= 



-1} = *(). 



.- 



294 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

The problem is to find the coefficients A(o>) and B(o>) such that these 
two conditions are satisfied, and thereby to show that the prescription 
of the function and its derivative at a specified point in space (or time) 
is sufficient to determine \l/(z, t) everywhere. 

If we assume provisionally that the integral on the right of (44) is 
uniformly convergent, it may be differentiated under the sign of integra- 
tion with respect to the parameter z, yielding 

(46) ^ = - f " <*[A(<*)* - JB()e"*V fal dw. 

OZ V J oo 

On placing z = in (44) and (46), we obtain 



(47) 



F(f) 



. / 00 

= - [A( 
J-- 



- ()]-*" d; 



upon Comparison with (21) and (22) it is immediately evident that the 
coefficients of the factor c" fa * in the integrands of (47) are Fourier 
transforms. 



(48) 



^ (A - B) = ~ f 

t; 4TT J - 



< d*. 



Solving these two relations simultaneously for A and B and substituting 1 
a for t as a variable of integration, we obtain 



(49) 



e** da, 
e da, 



and on substitution into (44) there results: 

(50) *fcO-;c f" * f" /()[ fa( " + !-0 + X-7-0 

TT3T J eo */ 



/*oO /* 8 TT/\ .W 

- L f <fo I ^M (6 V _ 

4TJ-00 J-co CO 



Now the first of these double integrals is the Fourier expansion of a 
function which on reference to (14) may be written down at once. In 

1 The variables a. and /3 appearing in the next few pages obviously have no connec- 
tion with the real and imaginary parts of k a + ift defined on p. 273. 



SBC. 5.8] GENERAL SOLUTION 295 

the second let us invert the order of integration. 

(51) w,.,^f(t-ty+f(t + ?) 

+ - r F(a)da (^ sina 6 i.(->*f. 
2*- J-co v ' J- w v w 

The imaginary part of the last integral vanishes because 

1 . co . JN 

- sm - z sm coux n 
co v x ' 

integrates to an even function. Therefore, 

/ero\ I COZ t w rt dw I . C02 , N d!o> 

(52) I sm e tw(a ~ = I sin cos co(a 

V ' J-oo t; CO J-oo V ^ ' 03 



By a slight modification & t&e conditions which defined the step function 
(23), it is easy to show that each of the last two integrals represents a 
function with a single discontinuity. In fact 

^ when p > 0, 

Zt 

/* .- 

(53) I dx = 0, when p = 0, 

jo x 

~> when p < 0. 

z z 
Hence (52) will vanish whenever the arguments a -\ t and a t 

z z 

are of the same sign. Changes of sign occur at a = t and a = t -\ 

The integral (52) has, therefore, a non vanishing value only in the interval 

z z 

t < a < t + -> where it is equal to TT. The general solution of (42) 

is now established in terms of the prescribed initial conditions. 

(54) *(M) = | 

Let us define a function &() by the relation 

(55) A(/J) = -y f F(a) da. 

Jo 



296 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

Then 

(56) ~ - fc'GS) - 

In place of (54) one may write 



When 2 = 0, it is evident that ^(0, reduces to f(t), and (d\f//dz) z ^ to 
- /&', or F(). It may also be verified by differentiation and substitution 

that (42) is satisfied by any function whose argument is of the form t - 

The replacement of the variable t in/(0 by t results in a displacement 

or propagation without distortion of the function to the right along the 
positive z-axis with a velocity v. The initial distribution therefore 
splits into two waves, the one traveling to the right and the other to 
the left. Upon these two waves are superposed the h waves, so chosen 
that at z = they annul one another but such that the derivative of \l/ 
assumes the prescribed value F(t). 

Prescription of the two functions f(t) and F(t) on the plane z = is 
sufficient to determine completely the electromagnetic field. Let us 
suppose, for example, that the electric vector of the plane wave is polarized 
along the z-axis. The Maxwell equations are then 



uZ ut uZ ut 

Let E x be represented by the function \l/(z, t) in either (54) or (57) ; then 
on differentiating with respect to both t and z we find expressions for 
dHy/dt and dH v /dz with the aid of (58) ; from these in turn it is a simple 
matter to deduce the expression for H y . 

(59) JBf.= 



These fields are clearly such that over the plane 2 = 0, 

E, = /(O. ~^ 

(60) (2 = 0) 



SBC. 5.9] DISSIPATIVE MEDIUM 297 

The electromagnetic field is, therefore, determined by the specification, 
as independent functions of time on the plane z = 0, of any pair in (60) 
containing both f(t) and F(t). Thus the electric field and its normal 
derivative may be prescribed, or the electric and magnetic fields without 
restriction on the derivatives. 

6.9. Dissipative Medium; Prescribed Distribution in Time. W$ 
pass on now to the more difficult problem of finding an electromagnetic 
field in a conducting medium which reduces to certain prescribed func- 
tions of the time on a plane z = constant. Again we shall allow \l/ to 
represent any rectangular component of an electromagnetic vector 
satisfying the equation 



It will be convenient in what follows to introduce two new constants. 
(62) 



. <r 
a = =, 6 = , 



k = \/AiW 2 + tjLurco = - \Ao 2 + 26a>i. 
a 

Note that in a dissipative medium a differs from the phase velocity 
v = c/n of a harmonic component. In this notation the wave equation 
assumes the form 



(63) -. 
or dt dz 2 

Particular solutions of (63) that are harmonic in time were investigated 
in Sec. 5.2 and found to be 

(64) ^ = (Ae** + Be-^e-'. 

Assuming the coefficients A and B as well as the complex quantity 
fc to be functions of the frequency, we construct a general solution by 
summation of the harmonic components. 

(65) $(z, t) = 

Again, let us suppose that the wave function and its normal derivative 
are prescribed functions of time over the plane z = constant. For 
convenience the origin of the reference system is located within the 
plane so that the constant is zero. 



(66) 

In physical problems the character of the function ^(2, <) is such that the 



298 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP, V 

infinite integral converges uniformly. Assuming this convergence, it 
may be differentiated with respect to the parameter z. 1 



d\l/ f * 
"te I * 

On the initial plane, (65) and (67) reduce to 



(68) 



Again the coefficients of e~' may be interpreted as the Fourier trans- 
forms of f(t) and F(0 respectively. 



(69) 

Solving for A and B: 

(70) 






*-/;. 



/()+**(*) 



E da, 



l da. 



When these values are reinserted into (65) one obtains, after inversion 
of the order of integration, the expression 



(71) 



i f 80 

J 



sin kz 



Thus far the analysis has not deviated formally from that of the 
problem in a nondissipative medium. At this point, however, we encoun- 
ter a difficulty due to the complexity of k and we shall have to digress 
momentarily to discuss the representation of the function sin kz/k as a 
definite integral. The basis of this representation is an integral due to 
Gegenbauer 2 which in the simpler case at hand reduces to 

I See for example Carslaw, "Introduction to the Theory of Fourier's Series and 
Integrals," Chaps. IV and VI, MacmUlan, 1921. 

WATSON, "A Treatise on the Theory of Bessel Functions/ 1 p. 379, Cambridge 
University Press, 1922. Equation (72) is a special case of (69), p. 411, derived in 
Sec. 7.7. 



Sac. 5.9] DISSIPATIVE MEDIUM 299 



(72) E^Jf = 1 f J Q ( U gi 

u * Jo 



Je(w sin < sin 0) denotes the zero-order Bessel function of argument 
u sin <t> sin 6. 

Let us make the following substitutions: 



(73) u = v 

where X and M are real or complex constants. 

u cos ^ = v I p + ~ \ u sin < = t ~ (X - /*), 
(74) 

cos ^ = -> sin ^ = ^/l ^ 
v \ t; 2 



sn 

t; 



In terms of these new parameters, (72) becomes 



sint; \/(p 




Observe that the function to the left of Eq. (75) is the Fourier transform 
of a function g(fi) defined as follows: 1 

00) = 0, when |j9| > t;, 
(76) 




when ||3| < v. 

Finally, let us assign to the parameters in this general formula the particu- 
lar values 

t; = -; p = 0>, X = 26i, JDt sa 0. 



The desired result is then 



a sin - \/w 2 + 26a>i 



(** 

i /. 

J-* 

** 2 



After this small excursion we return once more to Eq. (71). In the 
second integral on the right-hand side, which for convenience may be 
1 CAMPBELL and FOSTER, loc. cit., No. 872.2. 



300 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

denoted i/% introduce (77) and then invert the order of integration. 
* 

(78) *, = 2 J\ dp e-* [JL J ^ dco 

where 

(79) *(, 0) F() 

The double integral within the brackets is the Fourier expansion of the 
function <t>(t ft 0); consequently, 



(80) *, = 2 F(* - 0) Jo V* 2 - a 2 
or, upon a slight change of variable, 

(81) ^ 2 = 



To calculate the first integral to the right of Eq. (71), which shall be 
denoted ^i, we need only replace F(a) in 1^2 by f(a) and differentiate 
partially with respect to z. Since the limits in (81) are functions of z, 
this differentiation must be effected according to the formula 



(83) 



d/J. 

wave function which in a conducting medium reduces on the plane 
2 = to f(t) and whose normal derivative reduces to F(t) is determined every- 
where by the expression 



(84) 



a 



Sue. 5.101 DISSIPATIVE MEDIUM 301 

It is apparent from (84) that the character of the propagation is pro- 
foundly modified by the presence of even a slight conductivity. At 
2 = the integrals vanish due to the identity of the limits. The initial 
pulse f(f) then splits into two waves of half the initial amplitude as in the 
nondissipative case. These partial waves are propagated to the right 
and to the left with a velocity a which is necessarily independent of 
frequency, and also independent of the conductivity. They are further- 
more attenuated exponentially in the direction of propagation, as might 
have been anticipated. Note that the attenuation factor is that 
approached by a harmonic component in the limit as <7 2 /e 2 o> 2 oo . 

/OR\ b __ 

(85) a- 

which is identical with (53), page 277. The initial function is no longer 
propagated without change of form, for the integrals represent contribu- 
tions to the field persisting for an infinite time at points which have been 
traversed by the wave front. The wave now leaves in its wake a residue or 
tail which subsides exponentially with the time. The nature of these con- 
tributions will be clarified by the numerical example to be discussed in 
Sec. 5.11. 

5.10. Dissipative Medium; Prescribed Distribution in Space. The 
initial conditions are frequently prescribed in another fashion. Let us 
suppose that at the instant t = the field \l/ and its time derivative 
are specified as functions of the space coordinate z. 



(86) *(*, 0) = g(z), (^J t _ o = G(z). 

A harmonic analysis is called for in space rather than in time and a 
general solution may be constructed from particular solutions of the type 
discussed in Sec. 5.3. 

(87) ^ = e~*'(Ae* + 
where 



/p IT 

(88) g = ^ - |p = 

The constants a and 6 are defined as in Sec. 5.9 but k is now a real variable. 
To eliminate the common exponential factor it is convenient to take 

(89) * = (i, t)e-, 



302 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

The general solution is to be constructed by assuming the amplitudes 
A and B to depend on k and then summing over all positive and negative 
values of k. 

(90) t*(* f = f [A(k)e** + B(k)e-*]e ik ' dk. 
For the derivative, we have 

(91) S = f iq[A(k)e** - B(k)*-**\e*' dk. 
dt J oo 

The procedure is now essentially the same as that described in the fore- 
going paragraph. At t = 0, 



(92) 



g(z) = f" [A(k) + B(k)]e dk, 

J- 

G(z) = f (ip[A(k) - B(k)] - b[A(k) + B(k)]}e* dk, 

J-oo 



and the coefficients are readily determined by the Fourier transform 
theorem. 



(93) 

*(*) = iL 



Substitution of these values into (90) leads, after simple reductions, to 

(94) u(z, *)=^-| da g(a) \ cos qt e^< M > dk 
&K J - J - 



-- q 
sin at , 



cfib. 



In virtue of (75) the function - 



+ 
\ 



a' 



can be expressed 



as a definite integral. To the parameters in (75) are assigned the valuet 
t; = at, q = fc, X = 6/a, /i = ~6/a, 



Sac. 5.10] DISSIPATIVE MEDIUM 303 

whence it follows that 



sin at * 
(95) X 




Let us denote the three terms on the right-hand side of (94) by MI, 
and MS respectively. Then 



(96) u, - i J_^ 49 [1 J_^ dfc J 



where 
(97) 



By the Fourier integral theorem, the double integral within the brackets 
represents the expansion of <t>(z 0, 0), and consequently 



(98) us = ^ (?( - j8)/ Vl8 2 - a 2 < 2 dj9, 

or, after a change of variable, 



, _ 
(99) u* = ^^ G(j3) J V(* ~ 2 - a 2 i 2 dp. 

In like manner, one obtains 



_ 

(100) Mj = 2 _ o< ff (j8) J, V(z-W- aV) dp; 
from this ui is derived by differentiation with respect to t. 

(101) , = ]|0(a + of) + i g(z - at) 



+at 



wave function which in a conducting medium reduces at t = to g(z) 
and whose time derivative at the same instant is G(z) is determined everywhere 
by the expression 



(102) *(z, f) = e-" g(t +'at) + g(z - at) 



304 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

In its essential characteristics this solution does not differ markedly 
from that derived under the assumptions of Sec. 5.9. The initial field 
distribution along the z-axis splits into two waves which are propagated 
to the right and left respectively. If the conductivity is other than 
zero, a residue or tail remains after the passage of the wave front, and the 
field subsides exponentially with the time. 

5.11. Discussion of a Numerical Example. The physical significance 
of these results can be most easily elucidated in terms of a specific 
example. Fresh water is a convenient medium for such a discussion, 
for at radio frequencies it occupies a position midway between the good 
conductors and the good dielectrics. Let us assume the following 
constants: 

K m = 1, K e = 81, <r = 2 X 10~ 4 mho/meter. 
From these values we calculate 

a = | X 10 8 meters/sec., 6 = 1.4 X 10 5 sec.- 1 , 

- = 4.2 X 10~ 3 meter- 1 , 
a 

It is usually easier to visualize a distribution and propagation along 
an axis in space than along one in time; for this reason we shall consider 
(102) first. Let us suppose that at t = the field vanishes everywhere 
except within the region 25 < z < 25, where its amplitude is unity. 

g(z) = 1, when |z| < 25, 

g(z) = 0, when |z| > 25, 

G(z) = 0, for all values of z. 

In Fig. 51a %[g(z + at) + g(z at)} is plotted as a function of z for 
various values of t, and in Fig. 516 the same as a function of t for various 
values of z. These two figures represent the propagation of the initial 
pulse in time and space in a medium of zero conductivity. The function 
g(z at) vanishes if 

z < at - 25 or z > at + 25, 

or, if 

t < i (z - 25) or t> - (z + 25). 

Likewise the negative wave g(z + at) is zero at all points for which 
z < -at - 25 or z > -at + 25, 



or 



t < -i (25 + z) or t > - (25 - z). 



SEC. 5.11] 



DISCUSSION OF A NUMERICAL EXAMPLE 



305 



In Figs. 52a and 526 the same wave functions are multiplied by the 
attenuation factor e~". Nothing has been stated regarding the origin 



g(z+at) 

\ 


g(z-at) g(z+ati 

f \ 


2*0 

r 


rl 


t = 0.45 x 10 sec. 

h" r! 


2-15 meters 

7 




^3.75xlO~ 6 sec. 
-1.0 


z 125 meters 
-1.0 


n 


n n 


n 


. 


t - 8.25 x 10 sec. 
-1.0 


z- 275 meters 
-1.0 


n_^ , 


, , , , n ,n . . , 


, , , n, 


300 -200 -100 100 200 300 -10-8-6-4-2 2 4 6 8 10 
z in meters t in micro-seconds 
FIG. 51a. FIG. 516. 


e -bt 


-bt -bt 

f g(z-at) | g(z+at) 


% bt g(z-at) 


r 


r \ 


-1.0 


rl 


= 0.45xlO~ 6 sec. 

r: r! 


2 15 meters 


rn 


t =* 3.75 x 10 sec. 

n 

1 1 1 1 


z =125 meters 
-1.0 

n 


I i r i i i 


*=8.25xlO~* sec. ^ 
-1.0 

, i i i r i i . i i 


z a 275 meters 
-1.0 

i i i rri i 



100 

z in meters t in micro-seconds 

FIG. 52a. FIG. 626. 

of the field, but it is supposed that at some anterior time it was generated 
by a system of appropriately disposed sources from whence the field is 
propagated and attenuated in such a manner that at t = it is distributed 



306 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

as shown at the top of Fig. 52a. In the present instance it is clear that 
the wave started as two pulses at some value of t < 10 X 10~ 6 sec., 
the one at a distant point on the positive z-axis, propagated to the left, 
and the other at a point on the negative z-axis, propagated to the right. 
In the neighborhood of t = 0, z = 0, they meet, pass through one another 
with consequent reinforcement, and then continue on their way. Their 
subsequent progress is indicated at t = 0.45, t = 3.75, and t = 8.25 
microsec. The succession of events occurring at a fixed point in space is 
illustrated in the series of Fig. 526. Take, for example, z = 275 meters. 
At t = 9 microsec. the front of a negative wave arrives at this point, 
traveling toward smaller values of z. An interval of 1.5 sec. is now 
required for this wave to pass, in the course of which its amplitude is 
diminished exponentially. From t = 7.5 to t = 7.5 microsec. all is quiet ; 
then arrives the front of the positive wave with greatly reduced ampli- 
tude, for it has passed through unit value at the origin and traveled on to 
the positive point of observation. 

This simple history of a field must now be modified to take into 
account the contributions of the integral terms in Eq. (102). In most 
cases the evaluation of these terms necessitates numerical or mechanical 
processes of integration, but thanks to the simple functional form of 
g(z) in the present instance the integration can be carried through 
analytically for sufficiently small values of z and L From the theory of 
Bessel functions we take the following relations: 

T f \ _ -| ^ | % 3s | 

t/oW i 22 ' 22 . 42 ~~ 22 . 42 . g2 ~r'** 

*> = -,), 



Introducing the appropriate value of x leads to 



< 



An investigation of the convergence of these series indicates that in the 
present instance no serious error will be committed if we limit ourselves 
to the first two terms, provided that z does not exceed 300 meters. The 



SEC. 5.11] DISCUSSION OF A NUMERICAL EXAMPLE 307 

Bessel functions will, therefore, be represented approximately by 

, , 1 









These functions are now to be multiplied by g(fi) and integrated with 
respect to ]8. With regard to the limits we note the following. Suppose 
that t is positive. If z at > 25 or z + at < -25, the resultant field 
$(z, t) is zero. If at + 25 < z < at - 25, the partial fields g(z + at) 
and g(z erf) are zero but the integrals do not vanish, and this no matter 



-1.0 

A0 

-(Xo 

-0.6 




-200 -150 -100 -50 



_ 6 

= 3.75x10 sec. 



-02 /Residue 



50 
z in meters 
FIG. 53a. 




100 



150 



200 



how large t may become. A residual field lingers in the region previously 
traversed by wave pulses. If z + at > 25 and z at < 25, the limits of 
integration may be fixed at 25, for beyond these values g(fi) is zero. 
Such limits are, therefore, appropriate for the entire region between 
the pulses, at + 25 < z < at 25. 

The results of the integration are plotted in Figs. 53a and 536. The 
effective field \Kz, is indicated by a heavy black outline; the partial 





X 




-1.0 

e125 meters 
-0.8 

-0.6 
-0.4 


-0.2 


1CL ^H3 -6 -4 


-2 
t in mici 
FIG. 


2 4 6 8 10 
o-seconds 



fields expressed by the various terms of (102) are shown dotted. Figure 
53a illustrates the distribution along the z-axis at the instant t = 3.75 
microsec. We note the beginning of a deformation of the initially rec- 
tangular pulse, the top sloping forward and the residual tail trailing 



308 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

behind. As the wave progresses, the sharp front decreases in height, 
BO that tail and pulse eventually merge into a rounded contour. In 
Fig. 53& observations are made at the fixed point z = 125 meters. At 
times anterior to t = 3 microsec. a field existed here, a field which was 
wiped out by the passage of the negative pulse. At t = 3 microsec. the 
positive wave arrives, followed by its tail which persists after the passage 
of the pulse. 

The essential difference between the solutions represented in Eqs. 
(84) and (102) is due to an interchange of the roles of space and time. 
To illustrate the behavior of the field, let it be assumed that the initial 
time function f(f) in Eq, (84) is a rectangular pulse defined by 

/(O = 1, when \t\ < 1 X 10~ 6 sec., 

f(t) = 0, when \t\ > 1 X 10~ 6 sec., 

F(f) = 0, for all values of t. 

b b 

a? / \ * / \ 

The functions y/U + ~J and ^rf(t - |) are plotted in Figs. 54a 



and 546 at various values of z and t. In virtue of the definition of 

it is clear that the third term on the right-hand side of (84) contributes 



/u+!> f /<*-> f f(t + %) f /<*-> 






20 *-0 




. \ 


- 








2 <10 meters 


f=0.5xlO~ 6 


sec. 


r 


i w r 


-jl.0 




Fr 




4^ 




\\ 


\\\ \\ 


\\ 






2=100 meters 


*-2xlO~ 6 


sec. 




rl.O 


-1.0 




n 


n n 


n 




-i 


2=300 meters 


*-8xlO~ 6 


sec. 




1.0 

, i i i 1 r 1 i i 


-1.0 

i i i i r 


7- i 



*10 -8-6-4-2 2 4 6 8 10 -300 -200 -100 100 200 300 
t in micro-seconds z in meters 

FIQ. 64a. FIG. 54&. 

nothing if t - - > 1 or t + ~ < -1. If 1 - - < t < - - 1, the funo- 
d a a a 

tions/U + -J and/ It j vanish but the integral does not. In this 

interval, which lies between the pulses on the J-axis, the limits of integra- 



SEC. 5.12] THEORY OF LAPLACE TRANSFORMATION 



309 



tion are 1. In Figs. 55a and 556 are shown the deformations of an 
initially rectangular pulse at fixed points in time and space due to the 
tail of the wave. 





-0.8 
-0.6 
-0.4 

-0.2 r 


f = 3.75x10"* sec. 

n 


1 ' ' 1 ~L 1 JLl 1 t__ 


00 -250 -200 -150 -100 -50 





z in meters 



250 300 



FIG. 55a. 

When studying these figures it must be kept in mind that two pre- 
existing waves coming from sources at infinity have been assumed 
which superpose to give just the correct distribution at t = or z = 0. 









-1.0 

-0.8 
2-125 meters 
-0.6 

-0.4 




-0.2 


12 -10 -8 -6 -4 


-2 

t in mici 

FIG. 


2 4 6 8 10 12 

oseconds 

556. 



A single wave whose amplitude is zero for all values of t and z less than 
zero can be generated only by a source at the origin constituting a 
singularity in the field a problem we have not yet considered. 

5.12. Elementary Theory of the Laplace Transformation. Among 
the oldest and most important of the various methods devised for the 
solution of linear differential equations is that of the Laplace trans- 
formation. If in the equation 



(103) 



dw 



the dependent variable w is transformed by the relation 
(104) w(z) = J* u(t)e~* di, 

it will be found in many cases that u(t) satisfies a differential equation 
which is simpler than (103), and in fact that u(() is frequently an ele- 
mentary function whereas w(z) cannot, in general, be expressed directly 
in terms of elementary transcendentals. 1 When the coefficients p(z) 
and q(z) are functions of the independent variable z, it is necessary to 

1 INCB, "Ordinary Differential Equations," Chaps. VIII and XVIII, Longmans, 
london, 1927. 



310 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

choose the path of integration properly in the complex plane of t in 
order that (104) shall be a solution of (103) ; the independent solutions are 
then distinguished by the choice of path. The result is a representation 
of the particular solutions as contour integrals. If, however, the coeffi- 
cients p and q are constant, the equation has no other singularities than 
the essential one at infinity and the paths of integration can be chosen to 
coincide with the real axis. The solutions are then expressed in terms 
of infinite integrals intimately related, if not equivalent, to a Fourier 
integral representation. 

In recent years the application of the Laplace transformation to 
linear equations with constant coefficients has been considered with 
growing interest on the part of physicists and engineers, and its orderly 
and rigorous procedure appears to be rapidly replacing the quasi-empirical 
methods of the Heaviside operational calculus. Although the Laplace 
methods lead to no results unobtainable by direct application of Fourier 
integral analysis, they do offer certain definite advantages from the 
standpoint of convenience. They are particularly well adapted to the 
treatment of functions which do not vanish at + oo and which conse- 
quently fail to satisfy the condition of absolute convergence, and they 
lead in a simple and direct manner to the solution of an equation in terms 
of its initial conditions. On the other hand, they can be applied only to 
problems in which the field may be assumed to be zero for all negative 
values of the independent variable; in short, the Laplace integral is 
applicable to problems wherein all the future is of importance, but the 
past of no consequence. 

We shall approach the theory of the Laplace transform from the 
standpoint of the Fourier integral and suggest rudimentary proofs for 
the more important theorems. Let us consider a function f(f) which 
vanishes for all negative values of t. Then apart from a factor l/-\/2w 
the Fourier transform of f(t) is 

(105) F() 

provided only that the integral f |/(0| dt exists. In case/() does not 

/o 

vanish properly at infinity the integral fails to converge, but under 
certain circumstances absolute convergence can be restored by intro- 
ducing a factor e" 7 '. The Fourier transform of e~^f(t) is then 

(106) F(y + tw) - f Q " /(O*-^** dt. 
If there exists a real number 7, such that 

(107) lim f* \tr"f (t)\dt < oo, 

y_ oo JO 



SBC. 5.12] THEORY OF LAPLACE TRANSFORMATION 311 

then f(f) will be said to be transformable. The lower bound y a of all 
the 7*8 which satisfy (107) is called the abscissa of absolute convergence. 
Suppose, for example, that f(t) = e bt when t ^ and f(f) = when 
t < 0. Then f(f) is transformable in the sense of (106) and 7 > 6 = y a . 
On the other hand a function f(t) = e* is not transformable, for there 
exists no real number 7 leading to the convergence of (106). 

The inverse transformation follows from the reciprocal properties of 
the Fourier transforms, Eqs. (21) and (22). 



(108) f(t)e-T = o- F(y + iu)e itu do>, 

*7T J oo 

where 7 > 7 a . The function defined by the integral on the right is 
equal to f(f)e"** only for positive values of t, aui vanishes whenever 
t < 0. To give explicit expression to this fact we introduce the unit 
step function u(t) which is zero when t < and equal to unity when t > 0. 
The inverse transformation may thus be written 

(109) /(O u(t) = ^ 



If y a <> 0, one may pass to the limit as 7 after the integrations 
indicated in (106) or (109) have been effected and thus determine the 
Fourier transforms of functions not otherwise integrable. Such a step 
is unessential, and the Laplace transform contains this convergence 
factor implicitly. Let us introduce the complex variable 5 = 7 + ico. 
Then the Laplace transform of f(t), henceforth designated by the operator 



(110) L[/(0] = J[ /(*>-" dt = F(s), Re(s) > 7 , 

where Re(s) is the usual abbreviation for "the real part of s," and y a 
is determined in each case by the functional properties of /(). The 
inverse transformation iy represented by an integral taken a/long a path 
in the complex plane of s. 

(111) L->[F(s)) - -L. (" ' " F(s)e ds = /() u(0, 

6in J 7 -ao 

where 7 > y a - The Laplace transformation can be interpreted as a 
mapping of points lying on the positive real axis of t onto that portion 
of the complex plane of s which lies to the right of the abscissa y a . The 
domains of f(t) and its transform F(s) are indicated in Fig. 56. 

From the linearity of the operators it follows that if L[fi(t)] = Fi(s), 
FiW, then 



(112) 



312 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
If, furthermore, a is any parameter independent of t and s, then 
(113) L[af(t)] = aF(s). 

The application of the Laplace transform theory to the solution of 
differential equations is based on a theorem concerning the transform of 

a derivative. Assume that/(0 and its derivative ~nf(f) are transformable 

in the sense of (107), and that f(f) is continuous at t = 0. 
Then by partial integration 



f 

Jo 



dt 



-f 

JO 



hence, if L[f(f)] = F(s), it follows that 



(114) 



The importance of this theorem lies in the fact that it introduces the 
initial value /(O) of the f unction /(). Integrating partially a second time 



t -plane 




Domain of f <ft 



FIG. 56. A Laplace transformation effects a mapping of points on the positive real axis of 
t onto the shaded portion of the s-plane. 

leads to a theorem for the transform of a second derivative which involves 
not only the initial value /(O) but also the value of the derivative 



(115) 



Thus the transform of the dependent variable of a second-order differential 
equation is expressed in terms of the initial conditions. These results 
have no analogue in the theory of the Fourier transform. 

The "Faltung" or "folding" theorem is a particular application of 
9, well-known property of the Fourier integral. 1 Suppose again that 



1 See for example Bochner "Vorlesungen uber Fouriersche Integrate," and Wiener, 
"The Fourier Integral," p. 45, Cambridge University Press, 1933. 



SEC. 5.121 THEORY OF LAPLACE TRANSFORMATION 313 



= Fi(s), L[fz(t)] = F 2 (s), and that we wish to determine the 
Laplace transform of the product function fi(t) 



(116) 



f 

Jo 



^ /*Y* + * /* o 

= . <foF 2 (<r) fMe-^' dt, 

ATM J^-tao JO 

and, hence, the first theorem: 



(117) L[A(0 -/2] - fV* ~ er) ' F,(<r) rfcr, 

^T ' 

72 > Ta,, 



Of more freqiient use is a second Faltung theorem on the inverse trans- 
formation of a product of transforms. We wish to determine a function 
whose Laplace transform is Fi(s) F 2 (s). 



(118) 



-T) dr. 



Since the unit function vanishes for negative values of the argument, 
the upper limit of T must be t and hence 



JQ 

or 

(120) L f f /!(* - r) / 2 (r) drl = FiGO F 2 (). 

LJo J 

Geometrically, the product 7 y1 i(s) F 2 (s) represents the transform of an 
area constructed as follows: The function fi(r) is first folded over onto 
the negative half plane by replacing r by T, and then translated to the 
right by an amount t. The ordinate fi(t r) is next multiplied by 
/ 2 (r) dr and the area integrated from to t 



314 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

A set of operations applied repeatedly in the course of our earlier 
discussion of the Fourier integral may be referred to as translations. Let 
a be a parameter independent of s and t. 

(121) L[e?'f(t)] = JT /(0e-c> dt - F(s - a). 

Equation (121) is obviously valid for both positive and negative values 
of a. Next consider the transform of f(t a). 

(122) L[f(t - a)] = jT"/( - a)e- dt = e f'jtfe-" dr. 



Now it has been postulated throughout that the f unction /(J) shall vanish 
for all negative values of the argument and hence the lower limit a 
may be replaced by zero. As a consequence, we obtain 



(123) LUd- a)] = r-FU, 
Likewise, we may write 

(124) L[/ + a)] *(.), 



but one must note with care that this last result applies only to functions 
f(t) which by definition vanish for t < a. A replacement of t by t a or 
t + a represents a translation of f(i) a distance a to the right or left 
respectively. 

In addition to the foregoing fundamental theorems there exist a 
number of elementary but useful relations which may be deduced 
directly from (110). The following formulas are set down for con- 
venience and their proof is left to the reader. 

(125) 
(126) 
(127) L /(*) = F(<r) da; 

(128) 



(129) /(O) =limsF(s); 

8 > oo 

(130) lim/(<) = limP(). 

t > oo s 

Evaluation of the contour integrals that occur in the inverse trans- 
formations (111) is most easily effected by application of the theory of 
residues. The functions f(t) with which one has to deal are analytic 



SBC. 5.12] THEORY OF LAPLACE TRANSFORMATION 315 

functions of the complex variable t except at a finite number of poles. 
It will be recalled that in the neighborhood of such a point, say t = a, 
the function may be expanded in a series of the form 



where $() is analytic near and at a and, therefore, contains only positive 
powers of t a. The singularity is said to be a pole of order m if m i? 
a finite integer. If, however, an infinite number of negative powers is 
necessary for the representation of f(t) , the singularity is said to be essen- 
tial. If now the function f(t) is integrated about any closed contour C 
in the i-plane which encircles the pole a but no other singularity, it can be 
shown that all the terms in the expansion (131) vanish with the excep- 
tion of the one whose coefficient is &_i. The result is 

(132) f /() dt = 2art6_i. 

Jc 

The coefficient 6_i is called the residue at the pole. In case the contour 
encloses a number of poles, the value of the integral is equal to 2iri times 
the sum of the residues. In particular, if f(f) is analytic throughout the 
region bounded by the contour C, the integral vanishes. These results 
are a consequence of a fundamental theorem of function theory. If f(t) 
is analytic on and within the contour C, which is to say that it may be 
expanded in terms of positive powers of t only then the value of f(t) 
at any interior point a may be expressed as 

(133) /(a) = ^ 

This is known as Cauchy's theorem. 

If the function f(t) vanishes properly at infinity, an integral which is 
extended, as in (111), along an infinite line may be closed by a circuit of 
infinite radius and thus made subject to the theorems of the preceding 
paragraph. The restrictions that govern the application of the theory of 
residues to the evaluation of infinite integrals are specified in what is 
known as Jordan's lemma quoted as follows: 

II Q(z] uniformly with regard to arg z as \z\ > when ^ arg z ^ TT, 
and if Q(z) is analytic when both \z\ > c (a constant) and ^ arg z ^ TT, then 



lim f e m <*Q(z) dz = 0, 

p oo /* 



where F is a semicircle of radius p above the real axis with center at the origin. 1 

1 WHITTAKER and WATSON, loc. tit., p. 115. In Chaps. V and VI of this reference 
the reader will find all that is essential for the application of function theory to the 
solution of physical problems. 



316 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

As stated, this lemma applies to the evaluation of integrals along 
the real axis from oo to +00, but it may be modified to include 
integrations along an imaginary axis; by appropriate changes the circuit 
may be transferred from the half plane lying above the real axis to that 
below, or from the right half plane to the left. In case Q(z) has poles 
within the closed contour at distances \z\ < c from the origin, the value 
of the integral is equal not to zero but to the sum of the residues. 

The bearing of these theorems on the theory of the Laplace transform 
may be illustrated by two elementary examples. Consider first the step 
function defined by 

/(0=0, when*<0, 
/(O = 1, when t > 0. 

It will be noted in passing that (134) does not fall within the province of 

/* 00 

functions admitting a Fourier analysis, for the integral I [/()[ dt 
is nonconvergent. The Laplace transform of (134) is 

(135) L [/()] = f 1 e~ ai dt = i = F(s). 

Jo s 

We verify this result by applying the inverse transformation (111). 

< 136 > 



Now F(s) has a pole of first order at s = 0. The abscissa of absolute 
convergence is zero, whence it is necessary that Rc(s) > 0. The integra- 
tion is to be extended along any line parallel to and to the right of the 
imaginary axis. The integrand of (136) satisfies the conditions of 
Jordan's lemma and the path of integration may be closed by an infi- 
nite circuit. If t > 0, the path may be deformed to the left so that 
Re(s) > oo. The closed contour 7 ^00 to 7 + i<x> to oo + i<x> to 
oo ioo to 7 i< contains the pole at s = 0. The residue is unity, 
and hence by (132) the right-hand side of (136) is also unity for t > 0. 
Since F(s) is only defined in the right half plane, a question may arise as 
to the justice of extending the integration to the left half plane. One 
must be careful to distinguish between a function and the representation 
of that function in a restricted domain. 1/s is a representation of F(s) 
when Re(s) > 0, but in view of possible discontinuities in F(s) the two 
functions need not coincide at all when Re(s) < 0. Now along the line 
7 1*00 to 7 + ioo we are integrating the function 1/s; for purposes of 
integration we may make use of any of its analytic properties. The 
analytic continuation into the left half plane is of the function 1/s, and 
not of F(s). 



SEC. 5.12] 



THEORY OF LAPLACE TRANSFORMATION 



317 



When t < 0, the vanishing of the integrand may be assured by 
deforming the path to the right. Within the closed contour 7 i oo to 
y + i oo to co + i oo to oo i oo to 7 t oo there are no singularities and 
the right-hand side of (136) is zero. 

We have verified that the unit step function may be expressed 
analytically by 



(137) 



1 /*7 + , 

= 2^ I 

AM J-y-too 5 



7 > 0. 



It is interesting to apply the translation theorem (123) to this result, 
putting a = z/v. 



(138) 
or 

(139) 



[( -?)]- 



-JrK 



!.(-:> 



*. 



From the preceding discussion it is apparent that (139) vanishes when 
t - < ; and is equal to unity when t - > 0. In Figs. 57a and 576 

ult - J is represented as a function of t and z respectively. 



u(t-z/v) 



t'=z/v t 

utt-z/u)=0, t<z/v, 
u(t-z/v)~l, t>z/v. 
Fia. 57a. 



u(t-z/v) 



z'=vt 



Q, z>vt, 
l, z<vt. 



Fia. 576. 



As a second example let us imagine that a harmonic wave is switched 
on suddenly at t = and continues indefinitely. 



(140) 
(141) 



= I <r*'-< dt = J-r-i 

JO S + tO) 



when t < 0, 
when t > 0. 

Re(s) > 0. 



As in the previous case we can verify (141) by applying the inverse 
transformation 



(143 > 



0. 



318 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

If t > 0, the path is again deformed to the left. The enclosed pole is 
located at s = io>; on applying the Cauchy theorem we see that (142) 
is indeed equal to e''. On the other hand if t < 0, the path is deformed 
to the right and the integral vanishes. In general, 

f t z\ 0, when t - - < 0, 

^ A'"*) ' v ' 



1 . . as = _i' w (*--l . z 

T -ioo 5 -T- tco e \ / when t > 0. 

' t; 

5.13. Application of the Laplace Transform to Maxwell's Equations. 

Let E x = f(z, t) and H v = g(z, t) be the components of a plane electro- 
magnetic field. The functions /(z, t) and g(z, t) are assumed to be trans- 
formable in the sense of (107) and to vanish for all negative values of t; 
moreover, they are related to one another by the field equations 

f?/ i ^ = ^2 4- ^4- f=0 

dz M dt ' dz "* dt J 

Treating z as a parameter, the Laplace transform of this system with 
respect to t proves to be 



z, s) - M (z, 0) = 0, 

(145) *?(, s) 

^|p + *sF(z, s) + ff F(z, s) - e/0, 0) = 0. 

Upon elimination of G(z, s) an ordinary, inhomogeneous equation for 
F(z, s) is obtained in which s enters only as a parameter. 



(146) - (^ 

Since the derivative of g(z, t) with respect to z at the instant t = can 
be expressed in terms of /(z, 0) and (6//d)-o through (144), this last 
result is equivalent to 



(147) - Cue* 2 + Ar<OF = - ( M s + /nr)/(e, 0) ~ M * 

By /(z, 0) one means strictly the limit of f(z, t) as > 0. The functions 

(148) MZ) = lim /(*, 0, /,(*) = lim %^, 

<-*o *-o vf 

represent the initial state of the field and are assumed to be known. Upon 
writing A 2 = /xes 2 + juo-s, we obtain finally 



(149) - AF = - 



SEC. 5.13] APPLICATION OF THE LAPLACE TRANSFORM 319 

The general solution of (149) consists of a solution of the homo- 
geneous equation 

(150) ^ - h*F = 

containing two arbitrary constants, to which is added a particular solu- 
tion of (149). One can verify easily enough that (149) is in fact satisfied 

by 

e hz C 

(151) F(z, s) = Ae h * + Be~ ha + e~ h *Z(z, s) dz 

JUI\J J 

e h *Z(z, s) dz, 



e -h, C 

~~2/rJ 



in which h is that root of \/A* which is positive when h 2 is real and posi- 
tive. The constants A and B are to be determined such as to satisfy 
specified boundary conditions at the points z = Zi and z = z 2 . After 
A and B have been evaluated in terms of s, an inverse transformation 
leads from (151) back to a solution /(z, t) of (144) satisfying both 
initial and boundary conditions. 

To illustrate, consider first the elementary case in which the field is 
zero everywhere at the initial instant t = 0. Then Z = and we are 
concerned only with the solution of the homogeneous equation. Let us 
suppose that at z = there is a source switched on at the instant t = 
which sends out a wave in the positive z-direction and whose intensity 
at the origin is /(O, f) = fs(t). Thus the second boundary condition at 
z = z 2 is replaced in the present example by the stipulation that the field 
shall be propagated to the right along the z-axis. The transform of the 
boundary value is represented by 

(152) F(0, s) = Fs() = L[Mt)]. 

Since the wave is to travel to the right, the constant A must be placed 
equal to zero. This is clear when v = 0, for then h = s/a, and by the 
translation theorem of page 314 

(153) F(z, s) = F*(8)e~* = L [/, (t - ^], 

(154) /(z, =/(- y> (t ^ 0, z > 0). 

If the conductivity is not equal to zero, the solution is more difficult 
In this case we shall write ft 2 = ( s-y/M* + 5 ,J- <r j j^ and apply the 



320 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
integral 1 



'* 



Upon substitution of \/j3 2 + z 2 = at, this reduces to 
(156) IT = a I r * Vo (a Vs 2 - a 2 '*) r" <8, 

a 

where a = 1/A/M*, 6 = o"/ 2 ^ a defined on page 297. Finally, by dif- 
ferentiation we obtain 



e~*^ - a \ e~~ bt ~ J Q V^ 2 - aW J e- 8t d, 



(157) 



or 

--z -/ 

(158) e~ h * = e * e ** - oLtofo t)] 



where 0(2, i) is a function defined by 
(159) 0(2, 



0, when < t < -> 



= e -bt 1. j ( V^ 2 - a), 
02 \a / 



when * > - 



With this transform at our disposal, we now write 

b z 



(160) F(, s) = F,( 

To this we apply the translation theorem and the Faltung theorem of 
page 313, followed by the inverse transformation of the complete equation. 

(161) f(z, t) = .TV, ( - |) - a /. - T)^ I /o 

a 

or, with a minor change of variable, 

(162) /(*,9-e~ f 



- or* 



f 



1 WATSON, toe. c&, p. 416, (4). 



SEC. 5.14] DISPERSION IN DIELECTRICS 321 

This function, as the reader should verify, satisfies the wave equation in a 
dissipative medium and represents a wave traveling in the positive direc- 
tion which at z = reduces to / 3 (0 for all values of t > 0. 

DISPERSION 

5.14. Dispersion in Dielectrics. A pulse or "signal" of any specified 
initial form may be constructed by superposition of harmonic wave 
trains of infinite length and duration. The velocities with which the 
constant-phase surfaces of these component waves are propagated have 
been shown to depend on the parameters e, /*, and a. In particular, if 
the medium is nonconducting and the quantities e and ju are independent 
of the frequency of the applied field, the phase velocity proves to be 
constant and the signal is propagated without distortion. The presence 
of a conductivity, on the other hand, leads to a functional relation 
between the frequency and the phase velocity, as well as to attenuation. 
Consequently the harmonic components suffer relative displacements in 
phase in the direction of propagation and the signal arrives at a distant 
point in a modified and perhaps unrecognizable form. A medium in 
which the phase velocity is a function of the frequency is said to be 
dispersive. 

At sufficiently high frequencies a substance may exhibit dispersive 
properties even when the conductivity a due to free charges is wholly 
negligible. In dielectric media the phase velocity is related to the index 
of refraction n by v = c/n, where n = VMW At frequencies less than 
10 8 cycles/sec, the specific inductive capacities of most materials are 
substantially independent of the frequency, but they manifest a marked 
dependence on frequency within a range which often begins in the ultra- 
high-frequency radio region and extends into the infrared and beyond. 
Thus, while the refractive index of water at frequencies less than 10 8 
is about 9, it fluctuates at frequencies in the neighborhood of 10 10 
cycles/sec, and eventually drops to 1.32 in the infrared. Apart from 
solutions or crystals of ferromagnetic salts, the dispersive action of a 
nonconductor can be attributed wholly to a dependence of *<, on the 
frequency. 

All modern theories of dispersion take into account the molecular 
constitution of matter and treat the molecules as dynamical systems 
possessing natural free periods which are excited by the incident field. 
A simple mechanical model which led to a strikingly successful dispersion 
formula was proposed by Maxwell and independently by Sellmeyer. 
A further advance was accomplished by Lorentz who extended the 
theory of the medium as a fine-grained assembly of molecular oscillators 
and who was able to account at least qualitatively for a large number of 
electrical and optical phenomena. According to Lorentz, however, these 



322 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

molecular systems obeyed the laws of classical dynamics; it is now known 
that they are in fact governed by the more stringent principles of quantum 
mechanics. Following upon the rapid advances of our knowledge of 
molecular and atomic structure revisions were made in the dispersion 
theory which at present may be considered to be in a very satisfactory 
state. 

Both the classical and the quantum theories of dispersion undertake 
to calculate the displacement of charge from the center of gravity of an 
atomic system as a function of the frequency and intensity of the dis- 
turbing field. After a process of averaging over the atoms contained 
within an appropriately chosen volume element, one obtains an expression 
for the polarization of the medium; that is to say, the dipole moment per 
unit volume. The classical result corresponds closely in form to the 
quantum mechanical formula and leads in most cases to an adequate 
representation of the index of refraction as a function of frequency. We 
shall confine the discussion, therefore, to the case in which the electric 
polarization in the neighborhood of a resonance frequency can be 
expressed approximately by the real part of 1 

(1) P = -^ ^ :- coE. 

coo or ^co(7 

By the electric field intensity, we shall now understand the real part of 
the complex vector 

(2) E = Eo 6~ tw '. 

The constant a 2 is directly proportional to the number of oscillators per 
unit volume whose resonant frequency is co . The constant coo is related 
to this resonant frequency by 

(3) 5% = co? - ia 2 , 

such that co > co at sufficiently small densities of matter. The constant 
g takes account of dissipative, quasi-frictional forces introduced by 
collisions of the molecules. The constants co and g which characterize 
the molecules of a specific medium must be determined from experi- 
mental data. 

At sufficiently low incident frequencies co, the polarization P according 
to (1) approaches a constant value 

1 For the derivation of this result, see Lorentz, "Theory of Electrons," or any 
standard text on physical optics such as for example Born, "Optik," Springer, 1933, 
or Forsterling, "Lehrbuch der Optik," Hirzel, 1928. 



SEC. 5.14] DISPERSION IN DIELECTRICS 323 

and, since the specific inductive capacity is related to the polarization by 

(5) P = (K - l)e E, 

one may express K in terms of the molecular constants. 

(6) K = 1 + |- 

When, however, the incident frequency is increased, further neglect 
of the two remaining terms in the denominator becomes inadmissible. 
In that case we shall define by analogy a complex inductive capacity K* 
through either of the equations 

(7) P = (K' - l)e E, D = K' E, 
whence from (1) we obtain 

(8) . -!+.. a : . . 

600 or lug 

In terms of this complex parameter the Maxwell equations in a medium 
whose magnetic permeability is /-to are 

(9) V X E + MO ~ = 0, V X H - eo/ ~r = 0, 

ot ut 

as a consequence of which the rectangular components of the field vectors 
satisfy the wave equation 

/IA\ ^9 i / dV n 

(10; \ z \f/ eoMo* -TTJJ- = 0. 

A plane wave solution of (10) is represented by 

(11) \[/ = ^ O e ***""*"', 
where 

(12) k = - V7' = a + ip, 

c 

so that 

(13) K' = ~ (a + #). 

The wave is propagated with a velocity v = co/a = c/n, but a and the 
refractive index are now explicit functions of the frequency obtained by 
introducing (8) into (13). 

In gases and vapors the density of polarized molecules is so low that 
K! differs by a very small amount from unity. The constant a 2 is there- 
fore small, so that w differs by a negligible amount from the natural 



324 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

frequency coo and the root of K? can be obtained by retaining the first two 
terms of a binomial expansion. Thus 

(14) - (a + iff) ~ 1 + 2 



yw. | v^j j - * O 2 9 

co 2coo co 2 tcogr 

When the impressed frequency co is sufficiently low, the last two terms 
of the denominator may be neglected so that 



The index of refraction and consequently the phase velocity in this case 
are independent of the frequency; there is no dispersion. 

If the impressed frequency co is appreciable with respect to the 
resonant frequency co but does not approach it too closely, the damping 
term may still be neglected. |co 2 co 2 | ^>> cog, 

/*/>\ c i 

(16) _ = n = l 



'" x I O 2 9 

CO 2 COQ CO' 5 

The attenuation factor is zero and the medium is transparent, but the 
refractive index and the phase velocity are functions of the frequency. 
If co < co , n will be greater than unity and an increase in co leads to an 
increase in n and a decrease in v. If co > coo, the refractive index is less 
than unity but an increase in co still results in an increase in the numerical 
value of n. The dispersion in this case is said to be normal. 

Finally let co approach the resonance frequency co . Upon resolving 
(14) into its real and imaginary parts one obtains 

ac 1 a 2 co 2 ) co 2 



" co ~ 2" (c^WTW 2 ' 

B = < L 

P 2c (co 2 - co 2 ) 2 + coV 

In Fig. 58 are plotted the index of refraction and the absorption coeffi- 
cient jftc/co of a gas as functions of frequency. The absorption coefficient 
exhibits a rather sharp maximum at coo, so that in this region the medium 
is opaque to the wave. As co rises from values lying below co , the index 
n reaches a peak at coi and then falls off rapidly to a value less than unity 
at co 2 , whence it again increases with increasing co, eventually approaching 
unity. When as in the region coico 2 an increase of frequency leads to a 
decrease in refractive index and an increase of phase velocity, the dispersion 
is said conventionally to be anomalous. According to this definition 
dispersion introduced by the conductivity of the medium and discussed 
in the earlier sections of this chapter is anomalous. In fact it might 



SBC. 5.15] 



DISPERSION IN METALS 



325 




FIG. 58. Dispersion and absorption 
curves in the neighborhood of a re- 
sonance frequency. 



almost be said tuat the " anomalous" behavior is more common than the 
"normal." 

In the case of liquids and solids these results must be modified to some 
degree since the assumption of a small density of matter is no longer 
tenable, but the general form of the dispersion curves is not greatly 
affected. Discussion has been confined, moreover, to the behavior of the 
propagation constants in the neighbor- 
hood of a single resonance frequency. 
Actually a molecule is a complicated 
dynamical system possessing infinite 
series of natural frequencies, each affect- 
ing the reaction of the molecule to the 
incident field. The location of these 
natural periods cannot be determined 
by classical theory; by proper adjust- 
ment of constants to experimental data, 
an empirical dispersion formula can be 
set up, of which (8) is a typical term 
and which is found to satisfy the observ- 
ed data over an extensive range of fre- 
quencies. 

6.15. Dispersion in Metals. A surprisingly accurate theory of the 
optical properties of metals may be deduced from the following rather 
crude model. One imagines the fixed, positive ions of a metallic con- 
ductor to constitute a region of constant electrostatic potential. Within 
this region a cloud or " gas" of freely circulating electrons has attained 
statistical equilibrium, so that on the average the force acting on any 
particular conduction electron is zero, and apart from normal fluctuations 
the total charge within any volume element is zero. The application of 
an external field E results in a general drift of the free electrons in the 
direction of the field. The motion is opposed by constantly recurring 
collisions at the lattice points occupied by the heavy ions; the consequent 
transfer of momentum from the drifting electrons to the lattice points 
results, on the one hand, in thermal vibrations of the ions, on the other in a 
damping of the electron motion. The exact nature of the local force 
exerted on a conduction electron is a somewhat delicate question, but in 
so far as one can ignore the contribution of bound electrons there appears 
to be justification for the assumption that the effective field intensity ig 
equal to the macroscopic field E prevailing within the conductor. 

Let r be the displacement from the neutral position of a free charge e 
whose mass is m. The charge is acted on by an external force eE and its 

motion is opposed by a force "- wflf -57 proportional to the velocity which 



326 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP- V 

accounts empirically for the dissipative effect of collisions. The equation 
for the average motion of e is, therefore, 

(18) mg 
whose steady-state solution is 

(19) r = 



g ^co com 
If there are N free electrons in unit volume, the current density is 

g2 

(20) J = tf<4 = is. 
v y * dt g io) 

Then by analogy with the static conductivity defined by J = o-E, one 
is led to introduce a complex conductivity <r f which in virtue of (20) is 

P i 

N- 

(21) J - 2L. 

v ' - 



The propagation constant and attenuation of a plane wave are now 
obtained in the usual way from the relation fc 2 = jueco 2 + tVjuco. The 
resonance frequencies of the bound electrons of metallic atoms are known 
to lie far in the violet or ultraviolet region of the spectrum, so that in 
the visible red, the infrared and certainly at radio frequencies the induc- 
tive capacity of a metal can be safely assumed equal to 6 . Moreover 
if the conductor is nonferromagnetic, ju is also approximately equal to 
/io and 

(22) & 2 = (a + i/3) 2 = 

Upon replacing a' by (21) and separating into real and imaginary parts, 
one obtains 



2 _*2-^ fl Ne * 1 \ 

a P c 2 V m co 2 + 2 / 



Ne* 



c 2 we co co 2 + gf 2 

At sufficiently low frequencies the inertia force mf in Eq. (18) is 
negligible with respect to the viscous force mgi . Within this range the 
conductivity <r' defined by (21) reduces to a real and constant value, 

(24) </ - > 

v ' mg 



SEC. 5.16] PROPAGATION IN AN IONIZED ATMOSPHERE 327 

which must be identical with the static conductivity a. In a series of 
researches of fundamental value, it was demonstrated by Hagen and 
Rubens 1 that the values of conductivity measured under static conditions 
may be employed without appreciable error well into the infrared. At 
wave lengths shorter than about 25 X 10~ 4 cm., however, the con- 
ductivity exhibits a pronounced dependence on frequency and the 
observed dispersion can be accounted for roughly by the relations set 
down in (23). 

The sum and substance of these investigations is to show that dis- 
persion formulas expressing the dependence of the parameters c, ju, and <r 
on frequency permit the extension of classical electromagnetip theory 
far beyond the limits arbitrarily established in Chap. I. 

5.16. Propagation in an Ionized Atmosphere. Recent investigations 
of the propagation of radio waves in the ionized upper layers of the 
atmosphere have led to some very interesting problems. In these 
tenuous gases the mean free path of the electrons is exceedingly long and 
the damping factor consequently negligible, so that the apparent con- 
ductivity turns out to be purely imaginary, 

AM 

(25) a' = i , 

mo) 

whence 

22 2 

(26) n 2 = 

V ' 



CO 



An electromagnetic wave is propagated in an electron atmosphere with- 
out attenuation and with a phase velocity greater than that of light in 
free space. At frequencies less than a certain critical value determined 
by the relation 

A/> 2 

w * = - 

the index of refraction becomes imaginary, and it can be shown that at the 
boundary of such a medium the wave will be totally reflected. 

Actually the problem of propagation in the ionosphere is complicated 
by the presence of the earth's magnetic field. Let us suppose that a 
plane wave is propagated in the direction of the positive z-axis of a 
rectangular coordinate system. For the sake of example we shall 
assume only that there are N particles per unit volume of charge e and 
mass m in otherwise empty space. There is also present a static mag- 
netic field that may be resolved into a component in the direction of 
propagation and a component transverse to this axis. Only the longi- 

i Ann. Physik, 11, 873, 1903. See Chap. IX, p. 508. 



328 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

tudinal component, whose intensity will be denoted by H , leads to an 
effect of the first order. 

The equations of motion of a particle are 



(28) y 



* + IT 

e 



Now the ratio of the force exerted by the magnetic vector of the traveling 
wave to that exerted by the electric vector is equal to the ratio of the 
velocity of the charge to the velocity of light. The terms involving 
H x and H y in (28) may, therefore, be neglected so that the equations of 
motion reduce to the simpler system 



x - - E 4- * <Q - - E - - z - 

x - m &* + m y, y - m ^y m x > *-" 

Since the motion takes place entirely in the ?/-plane and since the 
vectors of the incident wave are likewise confined to this plane, complex 
quantities can be employed to advantage. Let us write 

(30) u = x + iy, E = E x + iE y , H - H x + iH v . 

Then the equations of motion are expressed by the single complex 
equation 

,_,. 
(31) 



while for the field equations one obtains 

dE . 3H . dH , . dE 
(32) ^ - tMo - dT = 0, -.,- + t e = 

in which the charge per unit volume times the average velocity has been 
introduced for current density. 

A solution must now be found for this simultaneous system of three 
equations. Let us try 



(33) E = Ae l '0"- w , H = Be****-"*, u = 

These are harmonic waves traveling in the positive direction with a 
propagation constant h; whereas the algebraic sign of the exponent is 
immaterial in the absence of an applied field H , its choice in the present 



SBC. 5.16] PROPAGATION IN AN IONIZED ATMOSPHERE 



329 



instance gives rise to two distinct solutions as we shall immediately see. 
For upon introducing (33) into (31) and (32) one obtains 



(34) 



hA + IMOB = 0, 
+ ihB + NewC = 0. 



In order that this homogeneous set of equations for the amplitudes 
A, B, C have other than a trivial solution, it is necessary that the deter- 
minant vanish. 



(35) 



= <> 



m 



ih 





Neu 



= 0. 



Expansion of this determinant leads to an equation for the propagation 
constant h. 

Ne* 

_T_9 

(36) 



w* 



m 



The phase velocity v = w/A, and consequently (36) is equal to the square 
of the index of refraction n. 

It turns out, therefore, that an electron atmosphere upon which is 
imposed a stationary magnetic field acts like an anisotropic, double- 
refracting crystal in that there are two modes of propagation with two 
distinct velocities. It is apparent, moreover, that by an appropriate 
choice of co the index of refraction of the medium with respect to one of 
these waves becomes infinite. 



Ne* 



(37) 



m 



Likewise, at another frequency the index of refraction 



(38) 



We 



co s + 



m 



reduces to zero. The remarkable optical properties of the Kennelly- 
Heaviside layers can be largely accounted for by these formulas. A 



330 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

linearly polarized wave entering the medium is resolved into a right- 
handed and a left-handed circularly polarized component, the one 
propagated with the velocity v + and the other with the velocity v^. 
Since the reflection occurring at the layer is determined by the index of 
refraction, the polarization of the wave returning to earth may be greatly 
altered and frequently contains a strong circularly polarized component 
which contributes to " fading" phenomena. 1 

VELOCITIES OF PROPAGATION 

6.17. Group Velocity. The concept of phase velocity applies only to 
fields which are periodic in space and which consequently represent wave 
trains of infinite duration. If a state of the medium is represented by 
the function 1^(2, 0> where 

(1) iK, = Ae*'-*", 

then the surfaces of constant state or phase are defined by 

(2) kz ut = constant, 

and these surfaces are propagated with the velocity 



A wave train of finite length, on the other hand, cannot be represented in 
the simple harmonic form (1), and the term phase velocity loses its 
precise significance. One speaks then rather loosely of the " velocity of 
light," or of the " wave-front velocity." The necessity of an exact 
formulation of the concept of wave velocity became apparent some years 
ago in connection with certain experiments fundamental to the theory 
of relativity, and the subject at present has an important bearing upon 
the problem of communication by means of short waves propagated 
along conductors. Since an electromagnetic field can never be com- 
pletely localized in either space or time, there must be an essential 
arbitrariness about every definition of velocity. The slope and height 
of a wave front vary in the course of its progress, and the concept of a 
" center of gravity" grows vague as the pulse diffuses. It will probably 
be of more practical importance to the reader to know what certain 
common terms do not mean, than to attach to them a too precise physical 
significance. 

1 A more complete account of propagation in ionized media with bibliography 
will be found in an article by Mimno, Rev. Mod. Phya., 9, 1-43, 1937. See also the 
"Ergebnisse der exakten Naturwissenschaften," Vol. 17, Springer, 1938. 



SEC. 5.17] 



GROUP VELOCITY 



331 



Next in importance to the idea of phase velocity is that of group 
velocity. Let us consider first the superposition of two harmonic waves 
which differ very slightly in frequency and wave number. 



(4) 



\l/i = cos (kz co), 

^2 = cos [(k + 8k)z - (o> 



The sum of these two is 

(5) * = *i + ^ 2 = 2 cos (z dk - t 



~ 



which is just the familiar expression for the phenomenon of " beats." 
The field oscillates at a frequency which differs negligibly from w, while 
its effective amplitude 



(6) 



A = 2 cos i(z dk t 



varies slowly between the sum of the amplitudes of the component waves 
and zero. As a result of constructive and destructive interference the 
field distribution along both time and space axes appears as a series of 






FIG. 59. Beats. 



periodically repeated "beats" or " groups" in the manner portrayed by 
Fig. 59. Now the surfaces over which the group amplitude A is constant 
are defined by the equation 



(7) 



z dk t 5co = constant, 



from which it follows that the groups themselves are propagated with 
the velocity 

< 8 > M - w 

The group velocity is determined by the ratio of the difference of fre- 
quency to the difference of wave number. If the medium is nondis- 

persive, dk = - So> and, hence, the group velocity coincides with the 

phase velocity v. In a dispersive medium their values are distinct. 

In the example just cited the superposition of two harmonic waves 
leads to a periodic series of groups. A single group or pulse of any desired 
shape may be constructed upon the theory of the Fourier integral by 



332 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

choosing the amplitude of the component harmonic waves as an appro- 
priate function of the frequency or wave number and integrating. 



(9) *(*, = w A (fc)e *'-<> dk. 

The concept of group velocity applies only to such Fourier representa- 
tions as are confined to a narrow band of the spectrum. If the amplitude 
function A(k) is of negligible magnitude outside the region 

fco dk <> k < fc + 8k, 
we may replace (9) by 



which represents what is commonly called a "wave packet/* For the 
present we shall assume that fc is real and that w is a known function of fc. 
Within a sufficiently small interval 26fc, o>(fc) will deviate but slightly from 
its value at fco and may, therefore, be represented by the first two terms of 
a Taylor series. 

(U) co(fc) = co(fco) + fe) (fc - fco) + . 

\a,K/ fc-fco 

The condition that higher order terms in this expansion shall be negligible 
imposes the necessary restriction on dk. 

(12) kz - orf == fcoz - o> J + (fc - fco) U - P~j J t\ + . 

A wave packet can now be represented by 

(13) t = ^ O e i(fco *- wo<) 
where ^o is a mean amplitude defined by 



.... . 

(14) ^o 

This amplitude is constant over surfaces defined by 

(15) z ( -JT ) * = constant, 
v ' \dk/ k = ko ' 

from which it is evident that the wave packet is propagated with the 
group velocity 



(16) "-V2S/* 

In case the medium is nondispersive, u coincides with the phase velocity 
v, but otherwise it is a function of the wave number fc . 



SBC. 5.18] WAVE FRONT AND SIGNAL VELOCITIES 333 

From the relations k = 27T/X, o> = kv, we may derive several equiva- 
lent expressions for the group velocity which are occasionally more 
convenient than (16). 

. _ _d_ / IA _ 4. fr J^ . _ \ .^ dv 1 



It is apparent from the manner in which the group velocity was 
defined that this concept is wholly precise only when the wave packet is 
composed of elementary waves lying within an infinitely narrow region 
of the spectrum. As the interval 5k is increased, the spread in phase 
velocity of the harmonic components in a dispersive medium becomes 
more marked; the packet is deformed rapidly, and the group velocity 
as a velocity of the whole loses its physical significance. Note that a 
concentration of the field in space does not imply a corresponding con- 
centration in the frequency or wave length spectrum, but rather the 
contrary. Consider, for example, the case of a harmonic wave train of 
finite length. 

/(z) = 0, when \z\ > -~> 

(18) I 

J(z) = cos fc 2, when \z\ < ~- 

Z 

The amplitude of the disturbance in the neighborhood of any wave 
number k is given by (40), page 292, 



and the form of this function is illustrated by Fig. 50. As the length L 
increases, the wave number interval dk within which A (k) has an appreci- 
able magnitude decreases. On the other hand a pulse produced by the 
abrupt switching on and off of a generator is diffused over the spectrum 
and cannot be represented even approximately by (10). This broad 
distribution over the spectrum is a common grief to every laboratory 
worker who must shield sensitive electrical apparatus from atmos- 
pheric disturbances or the surges resulting from the switching of heavy 
machinery. 

5.18. Wave Front and Signal Velocities. If the dispersion of the 
medium is normal and moderate, a pulse or wave packet may travel a 
great distance without appreciable diffusion; since the energy is presumed 
to be localized in the region occupied by the field, it is obvious that the 
velocity of energy propagation must equal at least approximately the 



334 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

group velocity. In a normally dispersive medium an increase of wave 
length results in an increase of phase velocity and hence by (17) the 
group velocity under these circumstances is always less than the phase 
velocity. If on the contrary the dispersion is anomalous as is the case 
in conducting media the derivative dv/d\ is negative and the group 
velocity is greater than the phase velocity. There is in fact no lack of 
examples to show that u may exceed the velocity c. Since at one time 
it was generally believed that the group velocity was necessarily equiva- 
lent to the velocity of energy propagation, examples of this sort were 
proposed in the first years following Einstein's publication of the special 
theory of relativity as definite contradictions to the postulate that a 
signal can never be transmitted with a velocity greater than c. The 
objection was answered and the entire problem clarified in 1914 by a 
beautiful investigation conducted by Sommerfeld and Brillouin, 1 which 
may still be read with profit. 

The particular problem considered by Sommerfeld is that of a signal 
which arrives or originates at the point z = at the instant t and 
continues indefinitely thereafter as a harmonic oscillation of frequency o>. 
The signal is propagated to the right, into a dispersive medium, and we 
wish to know the time required to penetrate a given distance. We have 
seen that the Fourier integral of a nonterminating wave train does not 
converge, but this difficulty may be circumvented by deforming the path 
of integration into the complex domain of the variable w. On the other 
hand, the theory of the Laplace transform was introduced for the very 
purpose of treating such functions; and indeed, it may be shown by a 
simple change of variable that the complex Fourier integral employed by 
Sommerfeld is exactly the transform defined in Eq. (143) of Sec. 5.12. 
It will facilitate the present discussion to modify the original analysis 
accordingly. 

At z = the signal is defined by the function 

1 f T+ia> e 7 0, when * < 0, 

(20) /<>0 = 2SJ 7 -,.. T+to* 8 -^*, when*>0. 

The representation of this field at any point z within the medium is 
obtained by extending (20) to a solution of the wave equation, which 
by (10), page 323, is 

* 2 /-i'!^-0 
d c 2 dt* ~ 

1 SOMMERFELD, Ann. Physik, 44, 177-202, 1914, BRILLOUIN, iUd., 203-240. A 
complete and more recent account of this work was published by Brillouin in the 
reports of the Congrte International d' filectricitt, Vol. II, l re Sec., Paris, 1932. 



SEC. 5.18] WAVE FRONT AND SIGNAL VELOCITIES 335 

Equation (21) is satisfied by elementary functions of the form 

exp ( st s ^ z J 
from which we construct our signal. 



, - 

J ^ ' 2m Jy-i* s + io) ' 

In a nondispersive medium, K! is constant and (22) is identical with 
(143), page 318. If, however, the medium is dispersive, K! is a function 
of s. We shall suppose that the dispersion can be represented by a 
formula such as (8), page 323, and upon replacing i& by s we obtain 



To include metallic conductors we need only put o> = 0. The wave 
function which for i > reduces to a harmonic oscillation at z = is, 
therefore, 

/ 7 4-too H*-/ 5 ^);;) 

(24) /(Z; Q 1 p + ^ / ; ^ 7 > 0. 

' v ' ' ' 



As s > ioo, jS(s) > 1. If now r = t -- < 0, the contour may 

c 

be closed, according to Jordan's lemma, page 315, by a semicircle to the 
right of infinite radius. This path excludes all singularities of the 
integrand and consequently f(z, t) = 0. Thus we have proved that at a 
point z within the medium the field is zero as long as t < z/c, and hence 
that the velocity of the wave front cannot exceed the constant c. 

If T = t -- > 0, the path may be closed only to the left. The 
c 

singularities thus encircled occur at the pole s = ia> and at the branch 
points of p. These last are located at the points where = and 
= oo . If /3s is written in the form 



f 9 K\ at o\ /s 2 + sg + o?g + a 2 

(*o; p(s) = ^/ 24! Z~^2 ' 

we see that 

= , when s = ^ ff H V4coJ - gr 2 , 

(26) J J 

= 0, when s = -^ gr g \/4(u 2 + a 2 ) - ^ 2 . 

The disposition of these singularities in the complex s-plane is indicated 



336 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
in Fig. 60a, where 

Now if one encircles a branch point in the plane of 5, one returns to the 
initial value of ft but with the opposite sign. This difficulty is obviated 
by introducing a "cut" or barrier along some line connecting a + and &+, 
and another between a_ and &_. Over this "cut" plane which in the 
function-theoretical sense represents one sheet of the Riemann surface 
of ft(s) the function ft(s) is single- valued; for, since it is forbidden to 
traverse a barrier, any closed contour must encircle an even number of 
branch points. The path of integration in (24), which follows the line 
drawn from 7 i oo to y + i <*> and is then closed by an infinite semi- 



s-plane 



"*" 1 

1 




i 







s-i 


lane 






7 


-fa 


-io) 




\l 


t 





a 6 

FIGS. 60a and 606. Paths of integration in the -plane. 

circle to the left, may now be deformed in any manner on the "cut" 
plane without altering the value of the integral, provided only that in 
the process of deformation the contour does not sweep across the pole at 
s = zw or either of the two barriers. In particular the path may be 
shrunk to the form indicated in Fig. 606. The contributions arising 
from a passage back and forth along the straight lines connecting Co 
and Ci and Co and C* cancel one another and (24) reduces to three 
integrals about the closed contours C , Ci, and C 2 . 
The first of these three can be evaluated at once. 



(28) /,(, i) = i 
whence by Cauchy's theorem 

(29) / (, = e~ 



s -r to) 



rrds, 



SEC. 5.18] WAVE FRONT AND SIGNAL VELOCITIES 337 

By (25) 



/o 
A / 
\ 



hence, 

(31) / (a, = e~^ z . e&~ 

The remaining two integrals which are looped about the barriers cannot 
be evaluated in any such simple fashion and they shall be designated by 



(32) 



r (<-/<);) 

i 2 (*,0 = ~ * . . ds. 

* JCi + C, S + tW 



Thus the resultant wave motion at any point within the medium can 
be represented by the sum of two terms, 

(33) f(z,t) =/ (M) +/(*,0- 

Physically these two components may be interpreted as forced and free 
vibrations of the charges that constitute the medium. The forced 
vibrations, defined by / (2, ), are undamped in time and have the same 
frequency as the impinging wave train. The free vibrations /i 2 (z, are 
damped in time as a result of the damping forces acting on the oscillating 
ions and their frequency is determined by the elastic binding forces. 
The course of the propagation into the medium can be traced as follows : 
Up to the instant t z/c, all is quiet. Even when the phase velocity 
v is greater than c, no wave reaches z earlier than t z/c. At t = z/c 
the integral /i2 (2, t) first exhibits a value other than zero, indicating that 
the ions have been set into oscillation. If by the term "wave front " we 
understand the very first arrival of the disturbance, then the wave front 
velocity is always equal to c, no matter what the medium. It may be shown, 
however, that at this first instant t = z/c the forced or steady-state term 
/o(z, just cancels the free or transient term/i 2 (z, t), so that the process 
starts always from zero amplitude. The steady state is then gradually 
built up as the transient dies out, quite in the same way that the sudden 
application of an alternating e.m.f. to an electrical network results in a 
transient surge which is eventually replaced by a harmonic oscillation. 

The arrival of the wave front and the role of the velocity v in adjusting 
the phase are illustrated in Fig. 61, which however shows only the steady- 
state term. The axis z/c is drawn normal to the axis of t The line at 
45 deg. then determines the wave-front velocity, for it passes through a 
point z at the instant z/c. The line whose slope is tan = c/v determines 
the time t = z/v of arrival at z of a wave whose velocity is v. Actually 
the phase velocity has nothing to do with the propagation; it gives 
only the arrangement of phases and, strictly speaking, this only in the case 



338 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 



of infinite wave trains. The phase of the forced oscillation is measured 
from the intersection with the dotted line at t z/v, and the phase at the 
wave front t = z/c is adjusted to fit. If v > c, < 45 deg. The phase 
of the steady state is again determined by the intersection with the line 
z/c at t = z/v, but the wave front itself arrives later. 

The transition from vanishingly small amplitudes at the wave front 
to the relatively large values of the signal have been carefully examined 




z/c c v 

FIG. 61. Determination of the phase in the steady state. 

by Brillouin. This investigation is of a much more delicate nature and 
we must content ourselves here with a statement of conclusions which 
may be drawn after the evaluation of (32). According to Brillouin a 
signal is a train of oscillations starting at a certain instant. In the course 
of its journey the signal is deformed. The main body of the signal is 



^ S\ 





FIG. 62a. Illustrating the variation in 
amplitude of the first precursor wave. 



Signal 



FIG. 626. Illustrating the arrival of 
the first and second precursors and the 
signal. 



preceded by a first forerunner, or precursor, which in all media travels 
with the velocity c. This first precursor arrives with zero amplitude, 
and then grows slowly both in period and in amplitude, as indicated 
in Fig. 62a; the amplitude subsequently decreases while the period 
approaches the natural period of the electrons. There appears now a new 
phase of the disturbance which may be called the second precursor, 

traveling with the velocity c >. c. This velocity is obtained by 

V ^0 + a 2 

assuming in the propagation factor P(iu>) of Eqs. (30) and (32) that the 
impressed frequency co is small compared to the atomic resonance fre- 
quency w . The period of the second precursor, at first very large, 



SBC. 5.18] 



WAVE FRONT AND SIGNAL VELOCITIES 



339 



decreases while the amplitude rises and then falls more or less in the 
manner of the first precursor. 

With a sudden rise of amplitude the main body, or principal part, of 
the disturbance arrives, traveling with a velocity w which Brillouin 
defines as the signal velocity. An explicit and simple expression for w 
cannot be given and its definition is associated somewhat arbitrarily 
with the method employed to evaluate the integral (32). Physically 
its meaning is quite clear. In Fig. 626 are shown the two precursors 
and the final sudden rise to the steady state. When restricted to this 
third and last phase of the disturbance the term " signal" represents that 
portion of the wave which actuates a measuring device. Under the 
assumed conditions a measurement should, in fact, indicate a velocity 




u = group velocity 
C/v, V ~ phase velocity 

c/w, w signal velocity 

FIG. 63. Behavior of the group, phase and signal velocities in the neighborhood of a 

resonance frequency. 

of propagation approximately equal to w. It will be noted, however, 
that as the sensitivity of the detector is increased, the measured velocity 
also increases, until in the limit of infinite sensitivity we should record 
the arrival of the front of the first precursor which travels always with 
the velocity c. Qualitatively at least we may imagine the medium as a 
region of free space densely infested with electrons. An infinitesimal 
amount of energy penetrates the empty spaces, as through a sieve, 
traveling of course with the velocity c. Each successive layer of charges 
is excited into oscillation by the primary wave and reradiates energy 
both forward and backward. By reason of the inertia of the charges 
these secondary oscillations lag behind the primary wave in phase, and 
this constant retardation as the process progresses through successive 
layers results in a reduced velocity of the main body of the disturbance. 
The wave-front velocity defined here is thus always equal to the con- 
stant c. The phase velocity v is associated only with steady states and 
may be either greater or less than c. The group velocity u differs from 
the phase velocity only in dispersive media. If the dispersion is normal, 



340 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

the group velocity is less than v; it is greater than the phase velocity 
when the dispersion is anomalous. In the neighborhood of an absorption 
band it may become infinite or even negative. The signal velocity w 
coincides with the group velocity in the region of normal dispersion, 
but deviates markedly from it whenever u behaves anomalously. The 
signal velocity is always less than c, but becomes somewhat difficult and 
arbitrary to define in the neighborhood of an absorption band. In 
Fig. 63 the velocities which characterize the propagation of a wave train 
in the neighborhood of an absorption band are plotted as ratios of c. 

Problems 

1. Let EQ be the amplitude of the electric vector in volts per meter and 3 the mean 
energy flow in watts per square meter of a plane wave propagated in free space. Show 
that 

3 - 1.327 X 10-' El watt /meter 2 , EQ = 27.45 A/S volts/meter. 

2. Let #o be the amplitude in volts per meter of the electric vector of a plane wave 
propagated in free space. Show that the amplitudes of the magnetic vectors HQ and 
BQ are related numerically to EQ by 

HQ =* 2.654 X 10~ 3 EQ ampere-turn/meter = J X 10~ 4 EQ oersted, 
B - } X 10~ 8 E Q weber /meter 2 = i X 10~ 4 EQ gauss, 

where the oersted is the unrationalized c.g.s. electromagnetic unit. If EQ is expressed 
in statvolts per centimeter, show that 

HQ = EQ oersted, BQ = EQ gauss. 

If a particle of charge q moves with a velocity v in the field of a plane electro- 
magnetic wave, show that the ratio of the forces exerted by the magnetic and electric 
components is of the order v/c. 

3. The theory of homogeneous, plane waves developed in Chap. V can be extended 
to inhomogeneous waves. If \l/ is any rectangular component of an electromagnetic 
vector, we may take as a general definition of a plane wave the expression 



in which ^ is a complex amplitude and < the complex phase. The propagation factor 
k is now a complex vector which we shall write as 

k = ki -f iks = ani + tfni, 

where ni and n 2 are unit vectors. Show that a is the real phase constant; that the 
surfaces of constant real phase are planes normal to the axis ni, and the surfaces of 
constant amplitude are planes normal to n 2 ; and that is the factor measuring attenua- 
tion in the direction of most rapid change of amplitude. The true phase velocity % 
is- hi the direction ni. 

Show that the operator V can be replaced by tk and the field equations are 

k X E - coB - 0, k B - 0, 

k X H + coD - -tj, k D - 0. 



PROBLEMS 341 

also that, if the medium is isotropic, 

fc* = k\ - k\ + 2tki k 2 = 6>V 



Let ki k 2 = kjct cos and find expressions for k\ and k* in terms of /* , *, and the 
angle 0. 

4. Continuing Problem 3, assume that the electric vector is linearly polarized. 
Show that E is perpendicular to both ki and k 2 and that H lies in the plane of these 
two vectors. What is the locus of the magnetic vector H? Inversely, show that if 
H is linearly polarized it is perpendicular to both ki and k 2 , while E lies in the ki, 
k^plane. These inhomogeneous plane waves are intimately related to the so-called 
"H waves" and "E waves," Sec. 9.18. 

6. The theory of plane waves can be extended to homogeneous, anisotropic 
dielectrics. Let <r 0, /* - MO, and the properties of the dielectric be specified by a 
tensor whose components are e/*. Choosing the principal axes as coordinate axes, 
we have 

DJ = eoKytf,-, 0* = 1, 2, 3). 

The field equations relating E and D are then 

k X (k X E) + o>VoD =0, k D = 0. 

Let k = fcn, where n is a unit real vector whose components along the principal axes 
are n\, n 2 , n^. Let v = w/fc and v, = C/\/KJ- The v } are the principal velocities. 
Show that the components of E satisfy the homogeneous system 



, =0, (j = 1, 2, 3), 

and write down the equation which determines the phase velocity in any direction 
fixed by the vector n. Show that this equation has three real roots, of which one is 
infinite and must be discarded. There are, therefore, two distinct modes of propaga- 
tion whose phase velocities are v' and v" in the direction n. Correspondingly there 
are two types of linear oscillation in distinct directions, characterized by the vectors 
E', D' and E 7 ', D". Show that these are related so that 

D' E" = D" E' = D' D" = 0, 
E' E" ?* 0, E n ^ 0. 

6. Show that in a homogeneous, anisotropic dielectric the phase velocity satisfies 
the Fresnel relation 

"-i n.* 

-+ " 

where the r? are the direction cosines of the wave normal n with respect to the prin- 
cipal axes, and the / are the phase velocities in thp directions of the principal axes. 
(See Problem 5.) Show also that 

3 




342 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

where the 7,- are the direction cosines of the vector D with respect to the principal 
axes. 

7. If in Problem 6 v\ > v z > v 3 , show that there are two directions in which v 
has only a single value. Find these directions in terms of Vi, t>2, t>s, and show that in 
each case the velocity is t> 2 . The directions defined thus are the optic axes of the 
medium which in this case is said to be biaxial. What are the necessary conditions 
in order that there be only one optic axis? 

8. Show that for a plane wave in an anisotropic dielectric 



S = E X H [n# 2 - (n E)E], 
Mo 2 

where v is the phase velocity defined in Problem 5. 

A vectorial velocity of energy propagation u in the direction of S is defined by the 
equation 

S = hu. 
The velocity u is related to the phase velocity v by 

t; == n u. 
Show that 

h = JE D + 5Mo# 2 

and, hence, that the energy flow is equal to the total energy density times the velocity 
u, with 



E*n - (n E)E 
v 




or in magnitude 



9. From the results of Problem 8 show that the magnitude of the velocity u of 
energy propagation in a plane wave can be found in terms of the constants 1/1, Vz t Va 
from the equation 

3 

^^S 



in which the lj are direction cosines of the vector u. 

Show also that there are, in general, two finite values of u corresponding to each 
direction. In what directions are these two roots identical? What is the relation of 
these preferred axes of u to the optic axes? 

10. A nonconducting medium of infinite extent is isotropic but its specific induc- 
tive capacity K = e/e is a function of position. Show first that the electric field 



PROBLEMS 343 

vector satisfies the equation 

V2T? I 1*2T? 
J2/ j /i JJ/ 



where k z - a> 2 /z and the time enters only through the factor exp( iut). 

Assume now that the spatial change of e per wave length is small, so that 
|VK|AI. Show that the right-hand term can be neglected and that the 
propagation is determined approximately by 

W + & = 0, 

where \l/ is a rectangular component of E or H and k is a slowly varying function of 
position. Next let 



where is the phase, fc = /c, and S a function of position. Show that the phase 
function S is determined by the equation 



where 



Note that S is in general complex, even though K is real, and that the amplitude of the 
wave is also a slowly varying function of position. 

11. The transition from wave optics to geometrical optics can be based on the 
results of Problem 10. Show that as the wave number k becomes very large, the 
phase function is determined by the first-order, second-degree equation 



4)' 



while the amplitude satisfies 

Vln 



Since S is now real, the wave fronts are represented by the family S = constant, and 
the wave normals or rays are given at each point by VS. The function S is called 
the "eikonal" and is identical with Hamilton's " characteristic function." Let the 
vector n represent the index of refraction \/K times a unit vector in the direction of 
the wave normal. Then 

VS = n, n V In ^ = - }V n. 

Note that these relations hold only in regions where the change in K per wave length is 
small. Hence they fail in the neighborhood of sharp edges or of bodies whose dimen- 
sions are of the order of a wave length. In such cases the complete wave equation 
must be applied. (Sommerfeld and llunge, Ann. Physik, 35, 290, 1911.) 



344 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 
Let dr be an element of length in the direction of a ray. Then 

S = J" n - dr. 

S is a function of the coordinates x , 2/0, 20 of a fixed initial point and the coordinates 
x, y, z of the terminal point reached by the ray. Of all possible paths through these 
two points, the ray actually follows that which makes S a minimum, or 



r( x >y> z ) 
I 

J(xo,yo,zo 



n - di = 0. 



This is Fermat's principle. If the disturbance originates at XQ, y$, Zo at the instant U. 
and arrives at x, y, z at t, then 

S(x , 2/0, z ; x, y, z) = t - to, 

and the path of the ray is such that the time of arrival is a minimum. 

12. Plane waves are set up in an inhomogeneous medium whose inductive capacity 
varies in the direction of propagation. Suppose this direction to be the z-axis and 
assume the field to be independent of x and y. The wave equation is then 

d 2 ^ 

j + *}(*)* = 0, 

where k\ = co 2 Mo and the time enters only in the factor exp( iwt). 

Suppose that K * KI as z > <, and K KI as z > <*>, where KI and K.I are con- 
stants. It is possible to construct four particular solutions behaving asymptotically 
as follows: 



^i exp(tfco V*i z), as z oo. 

\//2 exp( iko VKI z), as z oo. 

\^ 8 -* exp(tfc V KZ z), as z > oo . 

^ 4 > exp( t'fco v K2 z), as z > oo. 

Thus ^i and ^* reduce to plane waves traveling in the positive direction; ^2 arid ^4, 
to waves in the negative direction. Only two of these solutions are linearly inde- 
pendent and, hence, there must exist analytic connections of the form 

^8 = fll^l 4- O2^2, ^4 = ^1^1 + &2^2. 

Mathematically this represents the analytic continuation of ^g from one region to 
another, and physically the reflection coefficient is given by the ratio r = aa/ai. 
Let 



where = fcos/*, ics is a constant and s an additional parameter. Replace z by 
the new independent variable u = exp(f) and show that the wave equation reduces 
to hypergeometric form. Express ^i . . . ^4 as hypergeometric functions and cal- 
culate the joining factors and the reflection coefficient. The form of K(Z) is such as 
to admit a very wide choice in strata. (Epstein, Proc. Nat. Acad. Sci., 16, 627, 1930.) 
13. Ten times the common logarithm of the ratio of the initial to the terminal 
energy flow measures the attenuation of a plane wave in decibels Show that for a 



PROBLEMS 



345 



plane wave in a homogeneous, isotropic medium 

power loss = 8.686)3 



db/meter, 



where is the attenuation factor defined in Eq. (49), page 276. 

14. For the media whose constants are given below, plot the attenuation of a 
plane wave in decibels per meter against frequency from to 10 7 cycles /second. 
Use semilog paper or plot against the logarithm of the frequency. 



Material 


Conductivity, 
mhos/meter 


AO 




4 


81 


Fresh water . . 


10-' 


81 


Wet earth ... 


10" 3 


10 


Dry earth 


10 -5 


4 









16. A function /(a?) is defined by 



x > 0, > 0. 
x < 0, ft > 0. 



Show that 



2 ft cos ux 



16. Obtain a Fourier integral representation of the function f(l) defined by 



/(O 
f(t) 



0, 



t< 0, 

t > 0. 



Plot the amplitudes of the harmonic components as a function of frequency (spectral 
density) and discuss the relation between the breadth of the peak and the logarithmic 
decrement. 

17. The surges on transmission lines produced by lightning discharges are often 
simulated in electrical engineering practice by impulse generators developing a uni- 
directional impulse voltage of the form 

V = Fo - [e-<-0 - e 



where a and /3 are constants determined by the circuit parameters. This function 
represents a pulse or surge which rises sharply to a crest and then drops off more 
slowly in a long tail. The wave form can be characterized by two constants ti and It, 
where ti is the time required to reach the crest, and t* is the interval from zero voltage 
to that point on the tail at which the voltage is equal to half the crest voltage. A 
standard set of surges is represented by the constants 



(a) 



'0.5, 
.1.0, 
1.5, 



is -5, 

ti - 10, 

*, - 40, 



microseconds. 



From these values one may determine the constants a and 0. 



346 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

Make a spectral analysis of the wave form and determine the frequencies at which 
the spectral density is a maximum for the three standard cases. 
18. The "telegraphic equation " 



has the initial conditions 



Show that 



u = f(x), = g(x), when t = 0. 

at 



i c at d r 

u - \[f(x + at) +f(x - at)] + ~- \ f(x + 0) - 7 U 

^a J -at <M L 

H 

where 



~ a? 
a 



T COS (^ COS 



This solution was obtained independently by Heaviside and by Poincare*. 

19. An electromagnetic pulse is propagated in a homogeneous, isotropic medium 
whose constants are c, /x, a in the direction of the positive 2-axis. At the instant 
i = the form of the pulse is given by 

_J!l 
1 e 2& * 



where 5 is a parameter. Find an expression for the pulse f(z, I) at any subsequent 
instant. 

20. Wherever the displacement current is negligible with respect to the conduc- 
tion current (<r/eo> ^> 1), the field satisfies approximately the equations 

aH 

VXE+M -=0, VXH=J. 
dt 

Show that to the same approximation 

E=-f- H-SxA, 

dt n 

V 2 A - ^ =0, V A = 0, 

dt 
and that the current density J satisfies 

VJ - M' = 0, V-J=0. 



PROBLEMS 347 

What boundary conditions apply to the current density vector? These equations 
govern the distribution of current in metallic conductors at all frequencies in the radio 
spectrum or below. 

21. The equation obtained in Problem 20 for the current distribution in a metallic 
conductor is identical with that governing the diffusion of heat. Consider the one- 
dimensional case of a rectangular component / propagated along the 3-axis 

a 2 / a/ 

dz* *** dt 
Show that 



i C f" 

'2Yrf J_ 



/() 6 4 < d* 



is a solution such that at t = 0, / = f(z) , and which is continuous with continuous 
derivatives for all values of z when t > 0. 

22. In Problem 21 it was found that the equation 

av a^ 

v = 

az 2 dt 

is satisfied by 

*->fe 

From the theory of Fourier transforms show that this can be written in the form 

/T" ^7^ r 2 * Q g ~ CT + 2nir ) a 

* ~ \ ~i ^ I 6 4 "' ^ d< *' 

n =3 oo 

Obviously the equation is also satisfied by 



1 ^ri C 2 * 
=- > e^-^-^'/W 

"n^ooJo 



Compare these solutions and from the uniqueness theorem demonstrate the 
identity 



t 2 



This relation has been applied by Ewald to the theory of wave propagation in crystals. 

23. Show that if radio waves are propagated in an electron atmosphere in the 
earth's magnetic field, a resonance phenomenon may be expected near a wave length 
of about 212 meters. Marked selective absorption is actually observed in this region. 
Assume e = 1.60 X 10~ 19 coulomb /electron, m = 9 X 10~ 31 kg. /electron, and for the 
density of the magnetic field of the earth B Q = 0.5 X 10~ 4 weber /meter 2 . 

24. A radio wave is propagated in an ionized atmosphere. Calculate and discuss 
the group velocity as a function of frequency, assuming for the propagation constant 



348 PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA [CHAP. V 

the expression given in Eq. (36), page 329. The concentration of electrons in certain 
regions of the ionosphere is presumed to be of the order of 10 ls electrons /meter 3 . 

25. A plane, linearly polarized wave enters an electron atmosphere whose density 
is 10" electrons /meter 3 . A static magnetic field of intensity J5 = 0.5 X 10~ 4 
weber/meter* is applied in the direction of propagation. Obtain an expression 
representing the change in the state of polarization per wave length in the direction 
of propagation. 

26. A radio wave is propagated in an ionized atmosphere in the presence of a fixed 
magnetic field. Find the propagation factor for the case in which the static field is 
perpendicular to the direction of propagation. 

27. As a simple model of an atom one may assume a fixed, positive charge to 
which there is bound by a quasi-elastic force a negative charge e of mass m. Neg- 
lecting fractional forces, the equation of motion of such a system is 

mi +/r = 0, 

where r is the position vector of e and / is the binding constant. A static magnetic 
field is now applied. Show that there are two distinct frequencies of oscillation and 
that the corresponding motions represent circular vibrations in opposite directions 
in a plane perpendicular to the axis of the applied field. What is the frequency of 
rotation? This is the elementary theory of the Zeeman effect. 

28. The force per unit charge exerted on a particle moving with a velocity v in a 
magnetic field B is E 7 = v X B. The motional electromotive force about a closed 
circuit is 



V - w-as = [v X (v XB)]-ndd. 

At any fixed point in space the rate of change of B with the time is dB/dt. If by 
we now understand the total force per unit charge in a moving body, then 

V X E - ~ -f V X (v X B). 

dt 

Show that the right-hand side is the negative total derivative 

dB dB 

_=_. + (,. V)B , 

so that the Faraday law for a moving medium is 

dB 



The displacement velocity is assumed to be small relative to the velocity of light. 



CHAPTER VI 
CYLINDRICAL WAVES 

An electromagnetic field cannot, in general, be derived from a purely 
scalar function of space and time; as a consequence, the analysis of 
electromagnetic fields is inherently more difficult than the study of heat 
flow or the transmission of acoustic vibrations. The three-dimensional 
scalar wave equation is separable in 11 distinct coordinate systems, 1 
but complete solutions of the vector wave equation in a form directly 
applicable to the solution of boundary-value problems are at present 
known only for certain separable systems of cylindrical coordinates and 
for spherical coordinates. It will be shown that in such systems an 
electromagnetic field can be resolved into two partial fields, each derivable 
from a purely scalar function satisfying the wave equation. 

EQUATIONS OF A CYLINDRICAL FIELD 

6.1. Representation by Hertz Vectors. We shall suppose that one 
set of coordinate surfaces is formed by a family of cylinders whose ele- 
ments are all parallel to the z-axis. Unless specifically stated, these 
cylindrical surfaces are not necessarily circular, or even closed. With 
respect to any surface of the family the unit vectors ii, i 2 , is are ordered 
as shown in Fig. 64. ii is, therefore, normal to the cylinder, i 3 tangent 
to it and directed along its elements, and i 2 is tangent to the surface and 
perpendicular to ii and i 3 . Position with respect to axes determined 
by the three unit vectors is measured by the coordinates u l 3 u 2 , z, and 
the infinitesimal line element is 

(1) ds = ijii du l + i 2 h 2 du* + i 3 dz. 

Now let us calculate the components of the electromagnetic field 
associated with a Hertz vector n directed along the z-axis, so that 
H! = II 2 = 0, n r 7^ 0. We shall assume in the present chapter that the 
medium is not only isotropic and homogeneous but also unbounded in 
extent. Then by (63) and (64), page 32, the electric and magnetic 
vectors of the field are given by 



= \ ~ 



(2) E< = V X V X H, H< 1 > = \ ~ + er V X 

1 EISENHAHT, Annals of Math., 35, 284, 1934. 

349 



350 



CYLINDRICAL WAVES 



[CHAP. VI 



since both Hi and Hz are zero, it is a simple matter to calculate from (81) 
and (85), page 49, the components of E (1) and H (I) . 



(3) 



_ _., 



a 

- 



an. 



H< u = 0. 



Thus we have derived from a scator function IT Z = ^, an electromagnetic 
field characterized by the absence of an axial or longitudinal component 




y 

Fia. 64. Relation of unit vectors to a cylindrical surface. The generators of the surface 

are parallel to the 2-axis. 

of the magnetic vector. Since n is the electric polarization potential, this 
can be called a field of electric type (page 30), but in the present instance 
the term transverse magnetic field proposed recently by Schelkunoff l seems 
more apt. 

Since 11, is a rectangular component, it must satisfy the scalar wave 
equation, 

^2.f. 3J. 

(5) 



1 SCHELKUNOFF, Transmission Theory of Plane Electromagnetic Waves, Proa 
ln$t. Radio Engrs., 25, 1457-1492, November, 1937. 



SEC. 6.2] SCALAR AND VECTOR POTENTIALS 351 

or by (82), page 49, 

,-v 1 d /X d\l/\ 1 

(A) . I T ___L- I _L_ . 

W/ L 7, ^..1 I L 't-.l I r 7^ 7 



The elementary harmonic solutions of this equation are of the form 
(7) ^ = / n 2 )e ***-"**, 

where /(u 1 , w 2 ) is a solution of 

* 5 



In quite the same way a partial field can be derived from a second 
Hertz vector EL* by the operations 

(9) E< = -AI ^ V X n*, H< 2 > = VXVXIl*; 

(jt 

if n* is directed along the 2-axis, the components of these vectors are 

IL r) 2 TT* 

^ z 



(U) 

JJCD = _ A. [A ^ ^A + A Ai an?\l 

Aifca L^^ 1 \Ai 3u 1 / ^ dw 2 VA 2 du 2 / J 

The scalar function ILf is a solution of (5), and the field of magnetic type, 
or transverse electric, derived from it is characterized by the absence of a 
longitudinal component of E. 

The electromagnetic field obtained by superposing the partial fields 
derived from U 2 and II* is of such generality that one can satisfy a pre- 
scribed set of boundary conditions on any cylindrical surface whose gen- 
erating elements are parallel to the 2-axis; i.e., on any coordinate surface 
defined by u 1 = constant or prescribed conditions on either a surface 
of the family u 2 = constant or a plane z constant. The choice of these 
orthogonal families, however, is limited in practice to coordinate systems 
in which Eq. (8) is separable. 

6.2. Scalar and Vector Potentials. The transverse electric and trans- 
verse magnetic fields defined in the preceding paragraph have interesting 
properties which can be made clear by a study of the scalar and vector 
potentials. Consider first the transverse magnetic field in which 



352 CYLINDRICAL WAVES [CHAP. VI 

JJ(l) __ Q 

(12) E (1) = -V</> - ~^; B< = V X A, 

(13) 4>=-V.n, A = /ifej + ojn. 

\ x 

In the present instance T/' = II and, hence, 
d\f/ . d\fs 

(14) <f> == ) AZ AtC |~ P-^Yy A i A 2 == 0. 

The components of E (l) are then 



(15) 



for the components of B (1) , we obtain 




- o, 

in which A without a subscript has been written for A z . 

Now it will be noted that in the plane z = constant the vector E (1) 
is irrotational and therefore in this transverse plane, the line integral of 
E (1) between any two points a and 6 is independent of the path joining 
them; for an element of length in the plane z = constant is expressed by 
ds = ii/ii du l + i 2 A 2 du 2 and, consequently, 



(17) a B - ds = - du^ + d 

TAe difference of potential, or voltage, between any two points in a transverse 
plane has a definite value at each instant, whatever the frequency and whatever 
the nature of the cylindrical coordinates. 

Next, one will observe that the scalar function A plays the role of a 
stream or flow function for the vector B (1) . Let the curve joining the 
points a and 6 in Fig. 65 represent the trace of a cylindrical surface inter- 
secting the plane z = constant. We shall calculate the flux of the 
vector B (1) through the ribbonlike surface element bounded by the curve 
ab whose width in the ^-direction is unity. If n is the unit normal to this 
surface and is a unit vector directed along the z-axis, we have 



(18) B< 1 > n da = B< (i s X ds) = i 3 (ds X B<), 



SBC. 6.2] 



SCALAR AND VECTOR POTENTIALS 



353 



where ds is again an element of length along the curve. Upon expanding 
(18) we obtain 



(19) 

hence, 

(20) 



B< 1 > n da = 



du l 



du? = -dA; 
= A(a) - A(6). 




The magnetic flux through any unit strip of a cylindrical surface passing 
through two points in a plane z = constant 
is independent of the contour of the strip. 

If the components of E (1) and B (1) are 
expressed in terms of the scalar function ^, 
it is easy to show that 

(21) EyB<? + Ey*By> = o, 

and consequently the projection of the vector 
E (1) on the plane z constant is everywhere 
normal to B (1) . Moreover, in the trans- 

i xi_ r T f i 

verse plane the families of curves, <t> = con- 

Stant, A = constant, Constitute an 

orthogonal set of equipotentials and stream- 

lines. When the time variation is harmonic, it is clear from (14) and (7) 

that the equipotentials are the lines 



FIO. 65.- The curve a-b re- 

presents the intersection of a cy- 

ndrical 8urface with the xy _ plane 

which contains also the vectors n 



(22) 



/(te 1 , w 2 ) = constant, 



in which /(u 1 , u 2 ) satisfies Eq. (8). 

The transverse electric field has identical properties but with the 
roles of electric and magnetic vectors reversed. According to (35), 
page 27, 

(23) D< 2 > = -V X A*, H< 2 > = -V4>* - M! - ? A*. 

ut 

an* 

dt ' 



(24) <t>* = -V-n*, A* = 

If now we let t = n?, n? = II? = 0, then 



(25) <t>* = -^ A* == MC ^ Af = A? = 0. 

The components of the field vectors are, therefore, 

= o, 



d<t>* 



354 CYLINDRICAL WAVES [CHAP. VI 

(27) 

9V W 



The projection of H (2) on the transverse plane is irrotational and, con- 
sequently, the line integral representing the magnetomotive force 
between two points in this plane is independent of the path of integration. 

(28) 

Likewise the flux of the vector D (2) crossing a strip of unit width as shown 
in Fig. 65 depends only on the terminal points. 

(29) f b D< n da = f * dA* = A*(b) - A*(o). 

Ja Ja 

The projection of H (2) on the plane z = constant is everywhere normal 
to D (2) ; hence, the families </>* = constant, A* = constant are orthogonal. 
The field D (2) , H (2) is in this sense conjugate to the field <*>, B^. 

6.3. Impedances of Harmonic Cylindrical Fields. We shall suppose 
that the time variable enters only in the harmonic factor e~ itat . Then the 
potentials and field components of a transverse magnetic field are 



(30) <f> = +ih\[/, A = i(iJieo) + ii 

< 3I > w- w- * -<*>-, 



in which fc 2 = /zcco 2 + ip0w. The upper sign applies to waves traveling in 
the positive direction along the 2-axis, the lower sign to a negative direc- 
tion of propagation. 

A set of impedances relating the field vectors can now be defined on 
the basis of Sec. 5.6. It is apparent that the value of the impedance 
depends upon the direction in which it is measured. By definition, 

_^ _ By .COM/I 

" 



The intrinsic impedance of a homogeneous, isotropic medium for plane 
waves is by (83), page 283, 



(34) " */* + iff ~ k 



COjLt _ COjLt 



SEC. 6.4] ELEMENTARY WAVES 355 

so that (33) can be expressed as 

(35) Z< = |Z . 

Likewise impedances in the directions of the transverse axes can be 
defined by the relations 

(36) Ep = -ZpHp, E^ = ZpHp. 

The correlation of the components of electric and magnetic vectors and 
the algebraic sign is obviously determined by the positive direction of the 
Poynting vector. Since H ( z l) is zero, there is no component of flow 
represented by the terms E^H ( z l \ EpH ( , l \ and the associated impedances 
consequently are infinite. Upon introducing the appropriate expressions 
for the field components into (36), we obtain 

7CD _ W - /c 2 hrf 7 7 1} _ fe 2 -fc 2 h*t 7 

(37) ^ -- 3T~a /0 ' A -- ik~l& ' 

du l du 2 

The determination of corresponding impedances for the harmonic 
components of transverse electric fields needs no further explanation. 
The potentials and field vectors are in this case given by 

(38) 0* = +iht, A* = -i 



(40) Hi>= , ^'=' 

From these relations are calculated the various impedances. 



(42) 



(43) 

^ 



u^ 72) 

^o, Zi ^ 



There emerges from these results the rather curious set of relations : 
(44) Zi"Zi 2) = Zi"Z ( 2 2 > = Z( 1} Z^ 2) = Zi 2) . 

WAVE FUNCTIONS OF THE CIRCULAR CYLINDER 

6.4. Elementary Waves. By far the simplest of the separable cases 
is that in which the family u l = constant is represented by a set of 



356 CYLINDRICAL WAVES [CHAP. VI 

coaxial circular cylinders. Then by 1, page 51, 

(1) u l = r, w 2 = 0, A! =1, A 2 = r, 

and Eq. (8) of Sec. 6.1 reduces to 



which is readily separated by writing /(r, 0) as a product 

(3) / = /i(r)/,(0), 

wherein /i(r) and / 2 (0) are arbitrary solutions of the ordinary equations 



(4) r r + [(* - A')r - p 2 ]/, = 0. 

(5) + P% - 0. 

The parameter p is, like A, a separation constant; its choice is governed 
by the physical requirement that at a fixed point in space the field must 
be single-valued. If, as in the present chapter, inhomogeneities and 
discontinuities of the medium are excluded, the field is necessarily 
periodic with respect to and the value of p is limited to the 
integers n = 0, 1, 2, . . . . When, on the other hand, a field is 
represented by particular solutions of (4) and (5) in a sector of space 
bounded by the planes = 0i, = 0%, it is clear that nonintegral values 
must in general be assigned to p. 

Equation (4) satisfied by the radial function /i(r) will be recognized 
as Bessel's equation. Its solutions can properly be called Bessel func- 
tions but, since this term is usually reserved for that particular solution 
JpCv/fc 2 W r) which is finite on the axis r = 0, we shall adopt the name 
circular cylinder function to denote any particular solution of (4) and shall 
designate it by the letters /i = Z p (\/k* - h* r). The order of the func- 
tion whose argument is \/^ 2 h*r is p. Particular solutions of the 
wave equation (5), page 350, which are periodic in both t and 0, can there- 
fore be constructed from elementary waves of the form 

(6) ^. = e M Z n (Vk* - h 2 r) e**-H 

The propagation constant h is, in general, complex; consequently the field 
is not necessarily periodic along the z-axis. An explicit expression for h 
in terms of the frequency a? and the constants of the medium can be 
obtained only after the behavior of \l/ over a cylinder r = constant or on a 
plane z = constant has been prescribed. 



SUc. 6.5] PROPERTIES OF THE FUNCTIONS Z r (g) 357 

6.5. Properties of the Functions Z p (p). Although it must be assumed 
that the reader is familiar with a considerable part of the theory of Bessel's 
equation, a review of the essential properties of its solutions will prove 
useful for reference. 

If in Eq. (4) the independent variable be changed to p = \/k 2 h 2 r, 
we find that Z p (p) satisfies 



an equation characterized by a regular singularity at p = and an essen- 
tial singularity at p = oo. The Besself unction J p (p), or cylinder function 
of the first kind, is a particular solution of (7) which is finite at p = 0. 
The Bessel function can, therefore, be expanded in a series of ascending 
powers of p and, since there are no singularities in the complex plane of 
p other than the points p = and p = co , it is clear that this series must 
converge for all finite values of the argument. For any value of p, real 
or complex, and for both real and complex values of the argument p we 
have 



When p in (7) is replaced by p, the equation is unaltered; conse- 
quently, when p is not an integer, a second fundamental solution can be 
obtained from (8) by replacing p by p. If, however, p = n is an 
integer, J p (p) becomes a single-valued function of position. The gamma 
function T(n + m + 1) is replaced by the factorial (n + m)\, so that 

(9) J(P) = 



(n - 0, 1, 2, ... ). 

The function J- n (p) is now no longer independent of (9) but is related to 
it by 

(10) /-n(p) = (-l)J(p), 

so that one must resort to some other method in order to find a second 
solution. 

The Bessel function of the second kind is defined by the relation 

(11) N (p} = ihT^ [JM cos v* - / 

This solution of (7) is independent of J p (p) for all values of p, but the 
right-hand side assumes the indeterminate form zero over zero when p 



358 CYLINDRICAL WAVES [CHAP. VI 

is an integer. It can be evaluated, however, in the usual way by dif- 
ferentiating numerator and denominator with respect to p and then 
passing to the limit as p n. The resultant expansion is a complicated 
affair 1 of which we shall give only the first term valid in the neighborhood 
of the origin. 

(12) *,GO~- 



IT 7P 7T \p> 

(n = 1, 2, - ), 

where 7 = 1.78107 and [p| <JC 1. The characterizing property of the 
functions of the second kind is their singularity at the origin. Since they 
become infinite at p = 0, they cannot be employed to represent fields that 
are physically finite in this neighborhood. 

Further light is cast on the nature of the functions J p (p) and N p (p) 
when one examines their behavior for very large values of p. The expan- 
sions about the origin converge for all finite values of p, and J p (p) and 
Np(p) are analytic everywhere with respect to both p and p, the points 
p = and p = oo excepted. However, when p is very large the con- 
vergence is so slow as to render the series useless for practical calculation 
and one seeks representations of these same functions in series of 
inverse powers of p. It can, in fact, be shown that BessePs equation 
is satisfied formally by the expansions 



(13) JP(P) = \ [Pp(p) cos <t> Q P (p) sin $], 

/IF 

(14) N P ( P ) = ^~ [P p (p) sin * + Q p (p) cos 0], 



p p ( p ) = l - ^T ~ A ' w 9 ) 



2!(8p) 2 

- 25) (4p 2 - 49) 



KM <V> - 4p2 ~ 1 (4P 2 - l)(4p 2 - 9)(4p' - 25) 
(16) p (p) - ^ --- ^P - + 

in which the phase angle is given by 
(17)] * = p 



Now it turns out that these series diverge for all values of p, and 
consequently do not possess the exact analytical properties of the func- 

1 See WATSON, "A Treatise on the Theory of Bcssel Functions," Chap. Ill, 
Cambridge University Press, 1922, and JAHNKE-EMDB, "Tables of Functions/' 
p. 198, Teubner, 1933. 



SBC. 6.5] PROPERTIES OF THE FUNCTIONS Z 9 (g) 359 

tions they are intended to represent. On the other hand, if p is large, 
the first few terms diminish rapidly in magnitude and in this sense the 
series are "semiconvergent." It can be shown that if the expansions 
are broken off near or before the point at which the successive terms begin 
to grow larger, they lead to an approximate value of the function and the 
error incurred can be estimated. The larger the value of p the more 
closely does the sum of the first few terms coincide with the true value of 
the function ; for this reason such representations are said to be asymptotic. 
In the present instance we note that when p is sufficiently large, 



(18) J,(p) ~ - cos 



(p - 



At very great distances from the origin the cylinder functions of the first 
and second kinds are related to each other as the cosine and sine func- 
tions, but are attenuated with increasing p due to the factor l/\/P^ 
They are the proper functions for the representation of standing cylindrical 
waves. 

By analogy with the exponential functions one may construct a linear 
combination of the solutions J p (p) and N p (p) to obtain functions asso- 
ciated with traveling waves. The Bessel functions of the third kind, or 
Hankel functions as they are commonly known, are defined by the 
relations 



(20) H?(p) = J,(p) + tW,(p), 

(21) Hy>(p) = JM - iN,(p). 

From the preceding formulas we find readily that for very large p 
(22) 



("9" _/ _?! 
e V- 4 - 



(23) 

To the series expansions of the functions themselves we shall add for 
ready reference several of the more important recurrence relations. 

/94\ 7 I 7- ___ "P r7 

\^) 6p-l "T "p+l ^p. 

P 



360 CYLINDRICAL WAVES [CHAP- VI 

(25) ^.1^-1*^ 

(26) ~ 

(27) ~ 

6.6. The Field of Circularly Cylindrical Wave Functions. Within a 
homogeneous, isotropic domain every electromagnetic field can be repre- 
sented by linear combinations of the elementary wave functions 



(28) tnhk = e M J n (Vk* - h* r) e ** 

(29) 



Of these (28) alone applies to finite domains including the axis r = 0; 
at great distances from the source, (29) must be employed, since by (22) 
it reduces asymptotically to a wave traveling radially outward. Each 
elementary wave is identified by the parameter triplet n, 7i, k. When 
n = 0, the field is symmetric about the axis; when h = 0, the propagation 
is purely radial and the field strictly two-dimensional. Functions such 
as (28) and (29) can be said to represent inhomogeneous plane waves. 
The planes of constant phase are propagated along the 2-axis with a 
velocity v = w/a, where a is the real part of h, but the amplitudes over 
these planes are functions of r and 6. Such waves can be established 
only by sources located at finite distances from the origin of the reference 
system, or in media marked by discontinuities. The plane waves studied 
in the preceding chapter are in a strict sense homogeneous, since the planes 
of constant phase are likewise planes of constant amplitude. They can 
exist only in infinite, homogeneous media, generated by sources that are 
infinitely remote. 

From the formulas developed in Sec. 6.3 one may calculate the 
impedances and the components of the field vectors in terms of a wave 
function \l/. We find 

-A 2 )r 

> 



(30) a " 



k* 

(31) Ep = ih ^ E, == 7 ff' 

(32) H?> - -- - g* H.< - S g, = Q. 

x p.u r av n& or 



SBC. 6.7] CONSTRUCTION FROM PLANE WAVE SOLUTIONS 361 

Likewise for the transverse electric field, 



__, = 

(3g) -fc'Z.Cp) Jb'-Wr 



(34) * 

(35) #<*> = tt Jt ff,< = f ^ H{ = (* - A*)*. 

When initial conditions are prescribed over given plane or cylindrical 
surfaces, a solution is constructed by superposition of elementary wave 
functions. For fixed values of co (or fc) and h, one obtains for the resultant 
field in cylindrical coordinates the equations 



(36) E, = - - 

E, = (* - 



H. = (fc _ 



where a n and 6 n are coefficients to be determined from initial conditions. 
The direction of propagation is positive or negative according to the 
sign of h. 

INTEGRAL REPRESENTATIONS OF WAVE FUNCTIONS 

6.7. Construction from Plane Wave Solutions. If f measures dis- 
tance along any axis whose direction with respect to a fixed reference 
system (x, y, z) is determined by a unit vector n, then the most elementary 
type of plane wave can be represented according to Sees. 5.1 to 5.6 by 



362 

the function 
(1) 



CYLINDRICAL WAVES 



[CHAP. VI 



in which the constants k and w are either real or complex. Let R be the 
radius vector drawn from the origin to a point of observation whose 
rectangular coordinates are a;, T/, z. The phase of the wave function at a 
given instant is then measured by 



(2) 



f = n R = 



+ n y y + n z z. 



The direction cosines n x , n V) n z of the vector n are best expressed in terms 
of the polar angles a and ft shown in Fig. 66. 



(3) n x = sin a cos ft, 

hence, 



n v = sin a sin ft, 



= cos a; 



(4) 



gik(x sin a cos /3-j-y sin a sin /3-f* cos a) iut 



As the parameters a and ft are varied, the axis of propagation can be 

oriented at will. With each direction of 
propagation one associates an amplitude 
g(a, ft) depending only on the angles a 
and ]3; since the field equations are in the 
present case linear, a solution can be con- 
structed by superposing plane waves, all 
of the same frequency but traveling in 
various directions, each with its appropri- 
ate amplitude. 1 




(5) t(x, y, z, t) = e~^ J da f dft g(a, ft) 

gik(x sin a cos /3-fy sin a sin /S-f-z cos a) 



FIG. 66. The phase of an ele- 

mentary plane wave is measured If the angles are real, the limits of inte- 

* f ^tt*S^ s ration for * are obviousl y and *; ? g<* 

A J&xed point of observation is lo- from to 2?r. But Slich a solution is 

cated by the vector E. mathematically by no means the most 

general, for (5) satisfies the wave equation for complex as well as real 
values of the parameters a and ft and we shall discover shortly that 
complex angles must in fact be included if we are to represent arbitrary 
fields by such an integral. 

When w is real, the wave function defined by (5) is harmonic in time. 
To represent fields whose time variation over a specified coordinate 

1 The general theory of such solutions of the wave equation has been discussed by 
Whittaker. See WHITTAKBB and WATSON, "Modern Analysis," 4th ed., Chap. 
XVIII, Cambridge University Press, 1927. 



SBC. 6.7] CONSTRUCTION FROM PLANE WAVE SOLUTIONS 363 

surface is more complex it will be necessary to sum or integrate (5) with 
respect to the parameter co. We shall define a vector propagation constant 
k = fcn whose rectangular components are 

(6) ki = k sin a cos 0, k 2 = k sin a sin , & 3 = k cos a, 
so that the elementary plane wave function can be written 

(7) ^ = e <k.R-u^ 

It follows upon introduction of (7) into the wave equation 



te" aiT a? " a? " a ' 

that the components of k must satisfy the single relation 
(9) kl + kl + kl = /*co 2 



and are otherwise completely arbitrary. Thus of the parameters fci, fca, 
& 3 , co, any three may be chosen arbitrarily whereupon the fourth is fixed 
by (9). 

Let us suppose then that over the plane 2 = the function ^ is 
prescribed. We shall say that \l/(x, y } 0, f) = f(x, y, t). The desired 
solution is 



(10) 

KX, 2/, , = (^J J ^ J M J 



in which fti, ^2 and co are real variables and A; 3 is a complex quantity deter- 
mined by 



(11) k\ = C0 2 jLt + WCO kl fcf. 

The amplitude function is to be such that 



(\ 3 /* oo /* oo"l /* oO 

^) J- oo J^ co J- oo ' (fcl ' 



If /(x, t/, and its first derivatives are piccewise continuous and abso- 
lutely integrable, then g(ki, k^ co) is its Fourier transform and is given by 

(13) g(k lt fc 2 , co) = (^ J_ ^ J_ ^ J_ ^ /(*, y, e-^+*^-0 ctedy c. 

When <r = 0, each harmonic component is propagated along the z-axis 
with a velocity v = co/fc 3 , but since kz = -\/wVc ~~ &i "" ^2 is not a linear 
combination of co, fci, and fc 2 , it is apparent that the initial disturbance 



384 CYLINDRICAL WAVES [CHAP. VI 

/fo 1/9 does not propagate itself without change of form even in the 
absence of a dissipative term. More precisely, there exists no general 

solution of (8) of the type / 

Equally one might prescribe the function \{/(x, y, z, t) throughout all 
space at an initial instant t = 0. Let us suppose, for example, that 
\l/(x, y, z, 0) = f(x, y } z). The field is to be represented by the multiple 
integral 

(14) 

X 1 \ I /** /* eo / oo 

,/.// n I M f\ -~- I ll I I ndf-t Ufa. lfn\ t>i(ki%"\'lwU"\~k& <*>0 f-llf . fJlfnrllf- 

\y\X,y,Z)l) I TT I III yWly K'2) frS} & ' a/t/i U71/2 WA/3, 

V 71 / J-J-J- 

in which fci, & 2 , & 3 are real variables and co is a complex quantity deter- 
mined from (9). 



(15) iw = 6 i 






where a = l/\/M, & = 0"/2e as defined on page 297. The amplitude or 
weighting function g(k\, & 2 , ka) is to be such that 

(16) 



If f(x, y y z) has the necessary analytic properties, the Fourier transform 
exists and is given by 

(17) 

g(ki y k2j k&) = ( ^ ) I I I /(> yy *) e ~ IX 2J/ d% dy dz. 

Only positive or outward waves have been considered in the foregoing 
and initial conditions have been imposed only upon the function ^ 
itself. If both ^ and a derivative with respect to one of the four variables 
are to satisfy specified conditions, it becomes necessary to include nega- 
tive as well as positive waves according to the methods described for the 
one-dimensional case in Sees. 5.8, 5.9, and 5.10. 

6.8. Integral Representations of the Functions Z n (e). In any system 
of cylindrical coordinates t* 1 , w 2 , z, the wave equation (6), page 351, is 
satisfied by 

(18) ^ = f(u\ u 2 ) e*'-', 

where h and co are real or complex constants. Following the notation of 
the preceding section h = k$ = k cos a and, since k = \/Mo 2 + ^AWCO, it 



SEC. 6.8] INTEGRAL REPRESENTATIONS 365 

follows that the angle a made by the directions of the component plane 
waves in (5) with the z-axis is also constant. In other words, the ele- 
mentary cylindrical wave function (18) can be resolved into homo- 
geneous plane waves whose directions form a circular cone about the 
z-axis, but the aperture of the cone is in the general case measured by a 
complex angle. 



(19) f(u l , u 2 ) = f 0(0) e ik sin <* cos c+y sin dp, 

in which x and y are to be expressed in terms of the cylindrical coordinates 
w 1 , u 2 . 

In the coordinates of the circular cylinder we have x = r cos 0, 
y = r sin 6, and, hence, 

(20) x cos ft + y sin = r cos (6 0). 

In the notation of the preceding paragraphs we have also 

(21) kr sin a = r \/3k 2 h 2 = p, 
so that 

(22) /(r, 6) = ff(/5) * t<p cos (9 - ffi dp. 



We now change the variable of integration in (22) from p to <f> = 0, 
and note that, since the equation for /(r, 0) is separable, it must be 
possible to express g(<j> + 6) as a product of two functions of one variable 
each. 

(23) 0(0) = g(<t> + 0) = 
hence, 

(24) /(r, 0) - /i(r)/ 2 (0) = 2 (0) J 

The angle function 2 (0) must obviously be some linear combination 
of the exponentials e ipd and e~ lp9 , and the amplitude or weighting factor 
0i(<) must be chosen so that the radial function /i(r) satisfies (4), 
page 356, which in terms of p is written 

(25) pJ ^ + p |i + (p2 _ p% = . 
Let us substitute into (25) the integral 

(26) /i(p) = f0i(* 



366 CYLINDRICAL WAVES [CHAP. VI 

Differentiating under the sign of integration, we have 



cos 



^1 I i cos <t> fifi(*) e* p COB * d*, j% = - J 
hence from (25) 
(28) f (p 2 sin 2 + *p cos <t> - P 2 )0i(4>) e* c08 * d< = 0. 

This equation is next transformed by an integration by parts. Equation 
(28) is evidently equivalent to 



(29) J [ jrifo) " C 8 * + P 2 0i(4>) e* " *] d* = 0. 

If P and Q are two functions of <, then 

d dp dQ 
~ 



this applied to (29) gives 
(31) J - Ui(0) ^| 

+ 



^ + p0 1 ) 6 *o.* c ty = 



The first of these two integrals can be made to vanish if the contour of 
integration C is so chosen that the differential has the same value at both 
the initial and terminal points; the second is zero if the integrand van- 
ishes. Therefore (26) is a solution of the Bessel equation if gi(<t>) satisfies 

(32) j + pVi = 

d<p 

and if the path C of integration is such that 



(33) 



(ip sin gW - ^J 



= 0. 



Equation (32) is evidently satisfied by e ip +. However, we shall find 
that, if /i(p) is to be identical with the cylinder functions Z p (p) defined 

1 * T 
in Sec. 6.5, a constant factor - e ^ must be added. Then by choosing 

(34) "- 



SEC. 6.8] INTEGRAL REPRESENTATIONS 367 

we obtain Sommerf eld's integral representation of the cylinder functions, 1 

~ tp i (* 

(35) Z p (p) = - - e t-( 

v Jc 

the contour C to be such that 

(36) (p sin <(> + p) <>'(pcos*+P0) | = Q. 

The distinction between the various particular solutions J P (p\ 
N p (p), Hp(p) is now reduced to a distinction in the contours of integration 
in the complex plane of 0. Consider first the most elementary case in 
which p = ft, an integer. Then clearly, if the path of integration is 
extended along the real axis of </> from TT to TT, or an}?" other segment of 
length 2?r, the condition (36) is fulfilled and the definite integral 

' n 

(37) , J n (p) = ^ 

is a solution of the Bessel equation. That the function defined in (37) 
is actually identical with the particular solution (9), page 357, can be 
verified by expanding e ipcoa<f> in a series about the point p = and 
integrating term by term. 

In the general case when p is not an integer, or to obtain the inde- 
pendent solution, one must choose complex values of <j> in order that (36) 
shall vanish at the terminal points. Let = 7 + iy- Then 

(38) ip cos <t> + ip<t> = p sin 7 sinh 17 prj + ip cos 7 cosh 77 + ipy. 

If p is complex we shall assume that p = a + ib t where a is an essentially 
positive quantity. Then by an appropriate choice of 7 and t\ the real part 
of (38) can be made infinitely negative, while exp(tp cos <t> + ip<f>) 0. 
This condition will be satisfied, for example, if we let 77 > + and 
choose for 7 either ?r/2 or +37T/2; but the exponential also vanishes 
when 7i > oo provided 7 = 4-7T/2. In order that (35) shall represent 
a solution of the Bessel equation, it is only necessary that the contour C 
connect any two of these points. 

Sommerfeld has chosen as a pair of fundamental solutions the two 
integrals 

~ ip i /I"*" 

(39) #<,%>)=- e fr " *+fc* d0, 

* J--H- 

. T 

-"5 - 
(40) 



1 SOMMERFELD, Math. Ann., 47, 335, 1896. 



368 



CYLINDRICAL WAVES 



[CHAP. VI 



The contour C\ followed by the integral (39) starts at 17 = oo, 7 = ir/2, 
crosses both real and imaginary axes at <t> = 0, and terminates eventually 
at t\ = oo , 7 = ir/2. The contour C 2 followed by (40) starts at the 
terminal point of Ci, crosses the real axis at 7 = TT, and comes to an end 
at 17 = oo, 7 = 3?r/2. Both contours are illustrated in Fig. 67. The 
point at which the crossing of the real axis occurs is, of course, not essen- 
tial to the definition. The contours may be deformed at will provided only 
they begin at an infinitely remote central point of one shaded area and 
terminate at a corresponding point in a second shaded area. 




FIG. 67. Contours of integration for the cylinder functions. 

The advantages of carrying C\ and C 2 across the real axis at the 
particular points 7 = and 7 = TT become apparent when the integrals 
(39) and (40) are evaluated for very large values of p. For if the real 
part of p is very large, the factor exp(ip cos 0) becomes vanishingly 
small at all points of the shaded domains in Fig. 67 with the exception 
of the immediate neighborhood of the points r? = 0, 7 = 0, ir, 2ir, 
.... At these points the real part of ip cos <f>, Eq. (38), is zero however 
large p; consequently, if Ci and C 2 are drawn as in Fig. 67, the sole con- 
tribution to the contour integrals will be experienced in the neighborhood 
of the origin and the point rj = 0, 7 = TT. Out of a shaded region in 
which the values of the integrand are vanishingly small, the contour Ci 
leads over a steep "pass" or "saddle point " of high values at the origin, 
and then abruptly downwards into another shaded plane where the con- 
tributions to the integral are again of negligible amount. The contour 
C 2 encounters a similar saddle point at t\ = 0, 7 = TT. To confine the 
integration to as short a segment of the contour as possible, one must 
approach the pass by the line of steepest ascent, descending quickly from 



SEC. 6.9] FOURIER-BESSEL INTEGRALS 369 

the top into the valley beyond by the line of steepest descent. This 
means in the present case that the contours Ci, C* must cross the axis 
at an angle of 45 deg. The behavior of the integrals (39) and (40) in 
the neighborhood of a saddle point has been used by Debye 1 to calculate 
the asymptotic expansions of the functions H^(p) and H(p). When p 
is very large, both with respect to unity and to the order p t one obtains 
the relations (22) and (23) of page 359, thereby identifying the integrals 
(39) and (40) with the Hankel functions defined in Sec. 6.5. Debye 
considered also the case in which p was larger than the argument p, but his 
results have been improved and extended by more recent investigations. 
A contour integral representation of J P (p), when p is not an integer, 
follows directly from the relation 

(41) J p (p) = i[#S, 1} (p) + H^( P }\. 

ip-z / 

(42) J (p) ==: I * p CO8 *~t~ l P < (%(h 

2?T JCt 

The contour C$ shown in Fig. 67 represents a permissible deformation 
of Ci + C 2 . 

6.9. Fourier-Bessel Integrals. It has been shown how in cylindrical 
coordinates the two fundamental types of electromagnetic field can be 
derived from a scalar function $. In case the cylindrical coordinates are 
circular, \[/ is in general a function obtained by superposition of elemen- 
tary waves such as the particular solutions (28) and (29) of page 360. 
Our problem is now the following: at the instant t = the value of 
^(r, By z, t) is prescribed over the plane z 0; to determine ^ for all 
other values of z and t. 

Let us suppose then that when t = z = 0, \f/ = /(r, 6). We shall 
assume that /(r, 6) is a bounded, single-valued function of the variables 
and is, together with its first derivatives, piecewise continuous. Then 
/(r, 6) must be periodic in 6 and can be expanded in a Fourier series whose 
coefficients are functions of r alone. 

00 

(43) /(r, 0) = ^ /-W ein > /-W = i 

If now /(r, 0) vanishes as r oo in such a way as to ensure the con- 
vergence of the integrals J^ |/ n (r)| -\/rdr, then each coefficient f n (r) can 
be represented by a modified Fourier integral. 2 

1 DEBYE, Math. Ann. 67, 535, 1909. See also WATSON, Zoc. cit., pp. 235jf. 
1 The Fourier-Bessel integral ia established by more rigorous methods in Chap. XIV 
of Watson's "Bessel Functions." 



370 CYLINDRICAL WAVES [CHAP. VI 

Consider a function f(x, y) of two variables admitting the Fourier 
integral representation 

(44) 



, j,) = JL J_^ J 



A transformation to polar coordinates is now made both in coordinate 
space and in k space. 

% = r cos 0, y r sin 0, 
fci = X cos ft y /c 2 = X sin 0, 

so that in the notation of earlier paragraphs X = k sin a = \/k 2 A 2 . 
Then fcix + fc 2 y = Xr cos (ft 0) and (44), as a volume integral in 
k space, transforms to 



(46) /(r, 0) = ^ X d\ \ df) g(\, ft) e*' " ->. 

^* t/0 t/0 

The physical significance of this representation is worth noting. The 
function exp[iXr cos (ft 0) iut] represents a plane wave whose 
propagation constant is X, traveling in a direction which is normal to the 
2-axis and which makes an angle ft with the z-axis. Each plane wave is 
multiplied by an amplitude factor g(\, ft) and then summed first with 
respect to ft from to 2ir and then with respect to the propagation 
constant, or space frequency X. 
The transform of f(x, y) is 

(47) 0(fci, A*) =2 T j_ 

which when transformed to the polar coordinates f = p cos /*, 17 = p sin ju, 
leads to 



(48) jr(X, j8) = p dp dM /(P, M) 
Suppose finally that /(r, 0) = / n (r) e in5 . Then 

(49) 0(X, j8) = ^ f p dp / w (p) f " d/z e~ 

&K JO JO 

which upon a change of variable such as0 = Ai~"0 iris clearly equal to 

(50) 0(X, 18) = *W*> f " /n(p)/n(X P )p dp = 

Jo 



SBC. 6.10] REPRESENTATION OF A PLANE WAVE 371 

Likewise (46) becomes now 

(51) /(r, tf) = / n (r) e M = ^- f X dX g n (\) f ** dft <f>* <*-iO-Ktf H*) 

^ Jo Jo 

or upon placing < = 0, 

(52) /(r) e = e f n (X) J n (Xr)X dX. 

Jo 

Thus we obtain the pair of Fourier-Bessel transforms 

(53) /.(r) = I n (A)/ n (Xr)X dX, 

Jo 

(54) n (X) = f /(p)/ n (Xp)p dp. 

Jo 
The function 

(55) * = r "5! e<n * f \n(X)/ n (Xr)e*VF^T'*xdX 

nfHo J 

is a solution of the wave equation in circular cylindrical coordinates, 
which at t = reduces to /(r, 0) on the plane z = 0. But this obviously 
is not the most general solution satisfying these conditions, for we have 
still at our disposal the two parameters w and k which are subject only to 
the relation fc 2 = /-teco 2 + tcr/uo. If o) is real, harmonic wave functions 
such as (55) may be superposed to represent an arbitrary time variation 
of i// over the plane 2 = 0. If both positive and negative waves are 
considered, it is permissible also to assign values to both \l/ and d\l//dz 
at z = 0. The treatment of such a problem has been adequately illus- 
trated in Chap. V. 

6.10. Representation of a Plane Wave. The Fourier-Bessel theorem 
offers a very simple means of representing an elementary plane wave in 
terms of cylindrical wave functions. Let the wave whose propagation 
constant is k travel in a direction defined by the unit vector n, whose 
spherical polar angles with respect to a fixed reference system are a and ft 
as in Fig. 66, page 362. Then 

. ikz cos a fat 



and our problem is to represent the function 

fJ) f(x y) = e*'* Bin a(* coa 0-f-y aiu 0) = gikr sin a cos (ft 6) 

as a series of the form 

(58) f(r, 0) = /.(r) e. 



372 CYLINDRICAL WAVES 

By (43) we have 

1 P 2 * 

(59) f n ( r )= gt-fcrsinacostf-*)- 

&K Jo 
Upon replacing 6 by <, this becomes equal to 

(60) / n (r) = <? n (*~ ft )j n (kr sin a), 

and we obtain thereby the useful expansion 



[CHAP. VI 



(61) 



e ikr 8in cos V- *> = 



sn 



Several other well-known series expansions are a direct consequence 

of this result. Thus, if we put p = kr sin a, 0=0 ^ (61) 

z 

becomes 
(62) 

which upon separation into real and imaginary parts leads to 

00 

cos (p sin <) = 5) ^n(p) cos n0, 

n = oo 

(63) 

00 

sin (p sin <) = ^ ^n(p) sin n<i>. 

n = oo 

THE ADDITION THEOREM FOR CIRCULARLY CYLINDRICAL WAVES 

6.11. Several important relations pertaining to a translation of tho 
axis of propagation parallel to itself can be derived from the formulas of 

the foregoing section. In Fig. 68 
and Oi denote the origins of two 
rectangular reference systems. 
The sheet of the figure coincides 
with the zi/-plane of both systems 
and the axes Xi, y\, Zi through Oi are 
respectively parallel to x, y, z. The 
function J n (Xri) e in&l , when multi- 




FIQ. 68. Translation of the reference 

system. 



r , , , . ., . . 

P lied y exp(ihzi to>0, repre- 
sents an elementary cylindrical 
wave referred to the Zi-axis. We wish to express this cylindrical wave 
in terms of a sum of cylindrical wave functions referred to the parallel 
z-axis through 0. 



SBC. 6.11] ADDITION THEOREM 373 

We write first 
(1) /.(X 

From the figure it is clear that Q\ = + ^ and that 

/2\ fi cos \l/ = ri cos (0i 0) = r r cos (0 ), 

ri sin ^ ri sin (0i 0) = r sin (^ ). 

Moreover, in virtue of the periodic properties of the integrand, (1) is 
equivalent to 

(3) J n (Xn) e^ = f f e* >n CM <*-*>+ 

^*" J-1T 

It follows from (2) that 

(4) ri cos (<> i/O = r cos </> r cos (< 
hence, 

,_ x r /> x 1 f* Xrco8*-tXroCOB(0-ftf- ~ . 

(5) Jn(Xri) e in0 > = ^ l_ 6 \ 2 ' d<t>. 
By (61), page 372, we have 

(6) e -t\ro c 



and in virtue of the uniformity of convergence one may interchange 
the order of summation and integration. 

^mmmm 

t'Xr cos <-H(n-fm)( <t> s ) +tw(U 0o)-f in6 



Upon replacing B\ by 6 + ip this result becomes 
(8) /.(XrOe* 1 *- 



An analogous expansion for the wave function H } (\ri) e inei can be 
obtained by writing 



(9) 



//'"(xn) +* = 1 f e l>1 CM ( * 

^" JCi 



374 CYLINDRICAL WAVES [CHAP. VI 

where C\ is the contour described in Fig. 67, page 368, with a shift of 
amount ^ along the real axis to ensure the vanishing of the exponential 
at the terminals. Upon expanding the exponent by (2), we find 



(10) 



,/ 1 I 

in + = - I 
* JCi 



J 

2 '. 



If now |r| > |r cos (0 0o)|, one can proceed exactly as above and 
is led to the expansion 

en) 

If r = 0, the two centers coincide; J m (0) = for all values of m except 
zero, and Jo(0) = 1. The angle \l/ is zero and the right and left sides 
of (11) are clearly identical. The expansion can be verified also in the 
case when r and n are very large, for then the Hankel functions can be 
replaced by their asymptotic representation, (22), page 359. The 
angle ^ is approximately zero and r\ c^. r ~ r cos (0 ). The 
amplitude factor \/2/irri can be replaced by \^2/irr without appreciable 
error, but the term r cos (0 ) must be retained in the phase. Then 
(11) reduces to 

/ m (Xr ) 6 2 , 

and the right-hand side of (12) in turn is by (61), page 372, the correct 
expansion of the plane wave on the left. Indeed as r\ becomes infinite, 
the expanding cylindrical wave function designated by (11) must become 
asymptotic to a plane wave. 

When |r| < |r cos (0 0o)| the expansion (11) fails to converge and 
is replaced by 

CO 

(13) ff^(Xri) * in * = X #^Xro)/ n+ ,n(Xr) ^C^-*), 

which is finite at r = 0. At this point J r n +m(Xr) is zero unless m = n. 
Moreover ^~TT + and, since H ( ^(p) = e nvi H^(p)t the right 
and left sides of (13) are clearly identical. In the other limiting case of 
r very large, we write 7*1 = r r cos (0 ) and find by means of the 
asymptotic representations of the Hankel functions that (13) approaches 

(14) e -V cos (0-0o) = 



77 



a plane- wave function propagated from Oi towards along the line joining 
these two centers. 



SBC. 6.12] ELEMENTARY WAVES 375 

WAVE FUNCTIONS OF THE ELLIPTIC CYLINDER 

6.12. Elementary Waves. Circular wave functions are in a sense a 
degenerate form of elliptic wave functions obtained by placing the 
eccentricity of the cylinders equal to zero. The analysis of the field and 
the properties of the functions must inevitably prove more complex in 
elliptic than in circular coordinates, but the results are also fundamentally 
of greater interest. 

From 3, page 52, we take 

(1) u l - , u 2 = ry, { > 1, -Igi,^ 1, 

(2) Al== 



upon introducing these into (8), page 351, we find that /(, ??) must 
satisfy the equation 




This in turn is readily separated by writing / = /i()/ 2 (*?) and leads to 



(4) (? - 1) ' + { + [ejfti - W - &]/, = 0, 

(5) (1 - ) - , + (b - c* a (k* - A VIA = 0, 



where 6 is an arbitrary constant of separation. Thus /i() and / 2 (??) 
satisfy the same differential equation. Equations (4) and (5) are in fact 
special cases derived from the associated Mathieu equation 1 

(6) (1 - z 2 X' - 2(a + l)zw' + (6 - cW)w = 

by putting the parameter a = . These equations are characterized 
by an irregular singularity at infinity and regular singularities at z = 1. 
When :s> 1, (4) goes over to a Bessel equation. 

A certain simplification of (4) and (5) can be achieved by transforma- 
tion of the independent variables. Let 

(7) = cosh u, rj = cos v, 

so that the transformation to rectangular coordinates is expressed by 

(8) x == CD cosh u cos v, y = Co sinh u sin v. 

1 WHITTAKBR and WATSON, loc. cit., Chaps. X and XIX; INCE, " Ordinary Differ^ 
ential Equations," Chap. XX, Longmans, 1927. 



376 CYLINDRICAL WAVES [CHAP. V 

Then in place of (4) and (5), we obtain 

(9) 2~ + (c 2 X* cosh* u - 6)/i = 0, 

(10) Jr 2 + (6 - c 2 X 2 cos 2 t;)/, = 0, 

where as in the preceding sections X = \/k* h 2 . The distance between 
the focal points on the z-axis, Fig. 9, is 2c and the eccentricity of the 
confocal ellipses is e = 1/cosh u. The transition to the circular case 
follows as the limit when Co * and u o . Then Co cosh u > Co sinh u 
> r, and t; is clearly the angle made by the radius r with the x-axis. 
For this reason we shall refer to f\(u) as a radial function and / 2 (y) as an 
angular function. When u = 0, the eccentricity is one and the ellipse 
reduces to a line of length 2c joining the foci on the o>axis. 

The Mathieu equations (9) and (10) have been studied by many 
writers. We shall consider first the angular functions f*(v). There are, 
of course, solutions of (10) whatever the value of the separation constant 
6. But the electromagnetic field is a single-valued function of position 
and hence, if the properties of the medium are homogeneous with respect 
to the variable v y it is necessary that f%(v) be a periodic function of the 
angle v. Now Eq. (10) admits periodic solutions only for certain char- 
acteristic values of the parameter 6. These characteristic values form a 
denumerable set 61, 6 2 , . . . , 6 m , . . . . Their determination is a 
problem of some length, so we shall content ourselves here with a reference 
to tabulated results 1 and proceed directly to the definition of the functions 
satisfying (9) and (10) when b coincides with a characteristic value 6 m . 

For the proper values of 6, Eq. (10) admits both even and odd periodic 
solutions. The denumerable set of characteristic values leading to even 
solutions may be designated as 6^ and the associated characteristic 

1 A very readable account of the theory of the Mathieu functions has been given 
by Whittaker and Watson, loc. cit. y Chap. XIX; further details with extensive refer- 
ences to the literature have been published by Strutt in a monograph entitled "Lam6- 
sche- Mathieusche- und verwandte Funktionen in Physik und Technik," in the collec- 
tion "Ergebnisse der Mathematik," Springer, 1932. Tables of Mathieu functions 
and characteristic values have been published by Goldstein, Trans. Cambridge Phil. 
Soc., 23, 303-336, 1927. The functions Se m and So m defined in the text differ from 
the ce m and se m of these authors only by a proportionality factor. Whereas Goldstein 
chooses his coefficients such that the normalization factor is A/Tr, we have found it 
advantageous to normalize in such a way that the even function and the derivative of 
the odd function have unit value at the pole t; 0. See Stratton, Proc. Nat. Acad. 
Sci. U. S., 21, 51-56, 316-321, 1935; and MORSE, ibid., pp. 56-62. Extensive tables 
of the expansion coefficients D, the characteristic values b m and the normalization 
factors have been computed by P. M. Morse and will appear shortly. 



SBC. 6.12] ELEMENTARY WAVES 377 

functions can then be represented by the cosine series 

(11) Se m (c<>\ cos v) = ]' D cos nv, (m = 1, 2, 3, ), 



where the primed summation is to be extended over even values of n 
if m is even and over odd values of n if m is odd. The recursion formula 
connecting the coefficients D (coA) can be found by introducing (11) into 
(10). All coefficients of the series are thus referred to an initial one which 
is arbitrary. It is advantageous to choose this initial coefficient in such 
a way that the function itself has unit value when v = 0, corresponding to 
TJ cos v = 1. To this end we impose upon the Z> the condition 

(12) 

The odd periodic solutions of (10) arc associated with a second set of 
characteristic values to be designated by by. These functions can be 
represented by a* sine series 

(13) So m (co\, cos v) = ' F% s i n nv 



upon whose coefficients F(c A) we impose the condition V' nF = 1. 

n 

Consequently the derivative of So m (c Q \, cos t;) will have unit value at 
v = 0. 

(14) ^ So m (c Q \, cos w) =1. 

The characteristic functions Se m and So m constitute a complete 
orthogonal set. Let ty e) and ty e} be two characteristic values and Se lt 
Be, the associated functions. They satisfy the equations 

(15) + (6 J.) _ C 2 X 2 C0g 



(16) ~' + (&}> - c 2 X 2 cos 2 v)Se f = 0. 
Multiply (15) by Se h (16) by Set, and subtract one from the other. 

(17) TV ( Se > TV Sei ~ Sei TV Se <) + W ~ b ^ Se < Se >' = 0- 

Upon integrating (17) from to 2w and taking account of the periodicity 
of the functions, we obtain 



(18) Se^coX, cos v) Se,(eoX, cos ) dv = 



378 CYLINDRICAL WAVES [CHAP. VI 

The normalization factors N ( can be computed from the series expan- 
sions of the functions. 

In identical fashion one deduces that 



cos 5o y (c X, cos tO dv = > > 

JM 4 , l J. 

and 

(20) J Q * Sei(c Q \, cos v) So,(c X, cos t;) dv = 0, 

the last for i = j as well as i 7* j. 

With each characteristic value b ( there is associated one and only 
one periodic solution Se m . For this same number b ( there must exist 
however, an independent solution of (10). Since the second solution is 
nonperiodic it is unessential in physical problems so long as the medium 
is homogeneous with respect to the angle v. When, on the other hand, 
there are discontinuities in the properties of the medium across surfaces 
v = constant, the boundary conditions may require use of functions of 
the second kind. 

We turn our attention now to the radial functions. It can be shown 
without great difficulty that Eq. (9) is satisfied by an expansion in Bessel 
functions whose coefficients differ only by a factor from those of (11) 
and (13). Thus, when the parameter b assumes one of the characteristic 
values b ( , an associated radial function is 



(21) Rel(c Q \, Q 



where = cosh u y i m ~ n = exp i(m - n) | Since all solutions of 

Bessel's equation satisfy the recurrence relations (24) and (25), page 359, 
the even radial functions of the second kind are defined in terms of the 
Bessel functions of the second kind by 



(22) 

The convergence of such series of functions is not easy to demonstrate 
but appears to be satisfactory in the present instance. The great advan- 
tage of these representations is that they lead us at once to asymptotic 
expressions for very large values of c X. By (18) and (19), page 359, 
we have 

(23) Re^CoX, Q ~ -* cos ( Co X - ^t- 1 A 

VCoXf \ 4 / 

1 / o _L i \ 

(24) fle* M (c X, & ex -= sin ( Co X$ - ^Li A 



SBC. 6.12] ELEMENTARY WAVES 379 

By analogy with the Hankel functions, we are led to form the linear 
combinations 

(25) 



n 



(26) Re* m (c \, ) = Rei - iRe* m = 
whose asymptotic expansions are 

(27) lfci(coX, *) ^ - 



(28) 



The function Re^ satisfies the same equation as the angle function 
Se m ; neither has singularities other than that at infinity. The two must, 
therefore, be proportional to one another. 



(29) 

1 ^r-^/ 

--~> 

7(e) = 
m 



n - m -~ (meven), 



On the left, replaces cos v. Consequently when u = 0, = 1, we have 
(30) Bei(c X, 1) = -4=^)' [^ Bi(c X, cosh ti)l = 0. 

A corresponding set of relations can be obtained for the odd radial 
functions. We define: 

(31) 

(32) 

(33) BoJ^coX, ) = Ro^ 

The asymptotic expansions of the odd functions are identical with those 
of the corresponding even functions. Finally, 

(34) iSo m (co\, ) = V^S Qj^ifcoX, ), 

l p% in ~> (meven), 

~ i n ~ m , (m odd). 




380 CYLINDRICAL WAVES [CHAP. VI 

At u = 0, = 1, we have 

(35) Rol(c Q \ 1) = 0, [ - Rol(c,\, cosh u)l = ^ ~ 

L aM Ju0 V^TT^m 



With these functions at our disposal we are now in a position to write 
down elementary wave functions for the elliptic cylinder. The even and 
odd wave functions, which are finite everywhere and in particular along 
the axis joining the foci, are constructed from radial functions of the 
first kind. 



(36) fa = & w (c X, cos t>) Re l m (c,\ Q 

(37) fa = So M (coX, cos t>) #<4(c X, Q 



When it is known that at great distances from the axis of the cylinders the 
field is traveling radially outward, the elementary wave functions will be 
constructed from radial functions of the third kind. 



(38) fa = Se^CcoX, cos t;) Rel(c Q \, 

(39) fa = So m (coX, cos v) Boi(coX, 



The components of the electric and magnetic vectors can now be 
found by the rules set down in Sec. 6.3. 

6.13. Integral Representations. According to (19), page 365, we 
have in any system of cylindrical coordinates 



(40) f(u l , u*) = f 



e ik 8in 



If now the rectangular coordinates are replaced by elliptic coordinates 
through the transformation x = c cosh u cos v, y = Co sinh u sin v, we can 
write (40) in the form 

(41) f(u,v) = 

where as in the past X = \/k 2 h* = fc sin a, and where 

(42) p(u, v, 0) = x cos + y sin = Co(cosh u cos v cos 

+ sinh u sin v sin /3) , 
This function /(w, v) is to satisfy the equation 

(43) tot + + c X2 ( cosh2 w - cos2 )/ = 0, 
but since/ = /i(u)/ 2 (f), it is clear that it must also satisfy 

(44) f + (cgX 8 cosh 2 u-b)f = 0, 



SEC. 6.13] INTEGRAL REPRESENTATIONS 381 

obtained by multiplying (9), page 376, by / 2 (v). Upon differentiating 
(41) with respect to u, we have 



if, therefore, (41) is to satisfy (44), it is necessary that 

(46) J 0(0) [i\ ^ - X 2 feY + cy cosh 2 u-b\ e*" <Zj9 = 0. 

Now p is a function of j3 as well as of u and it can be easily verified that 



(47) d cosh* - = c? cos* /s + 

- - d * P - 

~ 



du* 
Consequently (46) is equivalent to 



Jf ft*p*f 1 

008) [-^r + (^ - CoX 2 cos 2 0)e*i>J d/S = 0. 

Finally by (30), page 366, 



hence, (41) is a solution of (44) provided g(f$) satisfies 
(51) + (6 - c 2 X 2 cos 2 fig = 0, 

and the path C of integration is so chosen that 



(52) 



= 0. 



It will be noted that the equation of the transform g(f) is identical with 
that of the angular f unction / 2 (v), and in fact differs unessentially from 
that for fi(u) with which we set out. This property is common to the 
transforms of all solutions of equations belonging to the group defined by 
(6), page 375. 

The results just derived are valid whatever the value of the separation 
constant b m . If, however, we confine ourselves to the even and odd sets 
of characteristic values 6J? and 6J?, it will be advantageous to choose for 
g(ff) the periodic solutions of (51) which have been designated by Se w (c X, 
cos /8), So m (c X, cos j8). Since p(u, v, ff) is also periodic in j3 with period 
2?r, it is apparent that (52) will be fulfilled when the path of integration 



382 CYLINDRICAL WAVES [CHAP. VI 

is any section of the real axis of length 2*. An integral representation 
of /(w, t;) which holds when b belongs to the set V is then 

/2r 

(53) f<?(u, tO = Se m (c Q \, cos 0) e W u >*> d|8. 

4/0 

From this an integral representing fi(u) can be found immediately by 

placing v = 0. 

^ 

(54) /ffi(w) = I /Se m (c X, cos |8) e* coX cosh u cos * djS. 

There remains the task of identifying the particular solution defined 
by (54). Upon introducing the cosine expansion of Se m and abbreviating 
coX cosh u p, we obtain 

(55) /fi(u) = 2' -" L eip C 8 * cos n/3 

n " 

Next we note that t n = (~l) n i~ n , and remember that the summation 
extends over even values of n if ra is even; over odd values only if w is 
odd. Therefore (~l) n = ( l) m , and in virtue of the definition (21), 
page 378, it follows that 

(56) ltei(coA, = ^~ C* Se m (c Q \ cos j3) e*** coa ^ dft 

V STT Jo 

where ^ = cosh u. Moreover, e m (c X, cos |9) is proportional to Re^(c Q \ 
cos 0), in which has been replaced by cos 0, so that by (29), page 379, 
the function Re^(c Q \j Q satisfies an integral equation, 

(57) Bei(c X, = *-* ^ f Re l m (c<>\ cos j8) e* 6 ^ C0fl ' dfr 



Still another representation of Re^ can be found by expanding the 
integrand of (57). Thus 



(58) Bi(coX, - 1^ ^ 2' ^^ J[ Jn(c X COS 

But 

*- r 

(59) J n (c X cos ]8) = cos n</> 6 l ' coX cos ^ cos 

T JO 

whence, 
(60) 

/2 / 

I / n (c X cos 0) e"' x * COB ^ d/9 = i~ n 

Jo jo 



cos n<^ c XJ + cos 
o 



SBC. 6.13] INTEGRAL REPRESENTATIONS 383 

Remembering once more that ( l) n = ( l) m , we obtain 

C f 2 * 
(61) Bei(coX, = ^(-l)" 1 A/O /o[coX(* + cos 0)] Se m (c X, cos *) <Z*. 



Integral representations of the radial functions of the third and 
fourth kinds can be obtained by a variation in the choice of contour. To 
simplify matters a little, let us take cosh u , v = 0, cos ft = t. Then 
the several radial functions are represented by integrals of the form 

(62) />(*) = J c Se m (c\ t) e***(l - * 2 )~* dt, 

where the contour C is such as to ensure the vanishing of the so-called 
"bilinear concomitant" (52), which now assumes the form 



= 0. 
c 



(63) (1 - **)*[ tCoX{& m (coX, - j t Se m (c*\, t) J e 

When t becomes very large, the asymptotic expansion of Se m is 

(64) Se m (c X, ^ W <~ cos 



consequently, the vanishing of (63) at infinity is governed by a factor 
exp icoX( l)t. If, therefore, the real part of c X( 1) is greater than 
zero, the contour can begin or end at t = i&> , while for the other limit one 
may choose either +1 or 1. As a result we have 



(65) Rel>(cQ\, ) = i m * /- I Se in (c Q \, t) < 

(66) .R0J(coX, ^) = i~ m ,* /- I Se m (cQ\j t) 



provided the real part of [c X( 1)] > 0. Reverting to the complex 
j3-plane, we find (65) equivalent to 



12 n"* 08 

Re* m (c \, Q = t-* J- J o 



(67) Re* m (c \, Q = t-* - o Sc w (c X, cos 

with a corresponding integral for jRe^ (c X, J). 

With the help of (67) one may deduce further integrals of the type 
(61). Thus, in place of (58) we write 

/|- 

(68) Bei(coX, Q = fi? \/2^ 2' '"" nD " J 

71 

' - !) nD n I d<t> cos n<^> I 2 ^+ C08 *> c09 ^ 



384 CYLINDRICAL WAVES [CHAP. VI 

Now by (39), page 367, when p = 0, we have 

HH" 1 f^'* 7T 

(69) I e ip cos ft d8 = 7: I e* p oos ft d8 = ^ H< Q l) (p), 

Jo 2 J_![ +ieo 2 

provided the real part of p > 0; consequently, 

(70) 

provided the real part of [c X( 1)] > 0. A combination of (61) and 

(70) leads to 



(71) #d(coX, )=Z^-l) m g o JV [c X(f + cos0)]Sc m (c X,cos^) d<t>. 

A clue to the formulation of integral representations of the odd func- 
tions is given by 

f 2 * n f 2 *" 

(72) e*** cos ^ sin np cos d/3 = -r-r-r e t>coX cos ^ cos n/5 d/5, 
v Jo ^c \f Jo 

the result of an integration by parts. The proof of the relations 

r \ _ /27T 

(73) Boi(c X, {) = -^ i 1 - 771 V? 2 - 1 ^o m (c X, cos/3) e* x * C08 ^sin/3dj3 

Jo 

^ 7 ! f Ro l m (c X, cos j8) e^ cos ^ sin /8 d/3 

Jo 

^ f " JJcoX( + cos )] 

Jo 

x .x sin , 

So m (c Q \ cos < 



- - , 

-p COS <p 



(74) Rol(c \, f) = Co X!>(- 1)- V=l ffi"[coX (f + cos 



is left to the reader. 

6.14. Expansion of Plane and Circular Waves. The periodic func- 
tions /Se w (coX, cos v) and So m (coX, cos v) constitute a complete, orthogonal 
set. Consequently, if a function f(u, v) is periodic in v with period 2ir and 
together with the first derivative df/dv is piecewise continuous with 
respect to t>, it can be expanded in a series of the form 



(75) /(u, v) = V /<J>M Se(c X, cos ) + j /Jj>(ti) So m ( Co X, cos ), 



SBC. 6.14] EXPANSION OF PLANE AND CIRCULAR WAVES 385 

whose coefficients are determined from 



(76) 



l C 2 * 

f^W = jj^ f(u, v) Se m (c<>\ cos v) dv, 

^ m t/0 

i f 2r 

/^M = ^53 J Q /(w, v) So m (c Q \, cos v) cto. 



This expansion may be applied to the representation of a plane wave 
whose propagation constant is A, traveling in a direction defined by the 
unit vector n whose spherical angles with respect to a fixed reference 
system are a and as in Fig. 66, page 362. 



(77) \l/ = e ik sin a ( x cos &+y sin ) e ijcz coa * w * 

Upon expressing x and y in terms of u and v and putting X = k sin a, we 
have 

(78) f(u, v, j8) = c^, p = Co (cosh w cos v cos + sinh u sin v sin 0). 

Since there is complete symmetry in (78) between v and j8, the expansion 
(75) can be written 



(79) e a " = 2 a w (w) & w (c X, cos /3) /Se m (c X, cos ) 

C^) So m (co\, cos 0) $o m (c X, cos v) 



in which the coefficients a m (u) and 6 OT (w) depend on u alone. In virtue 
of (76), we have 

(80) a m (u) NSe m (c<&, cos j8) = ** Se m (c Q \, cos v) e** dv. 



since a m (u] is unaffected by the value of 0, we may put 
Se m (co\ 1) = 1, p = c cosh M cos y. Then by (56), 



* -- 

(81) c^fy) = - ^ 0) Rel(c G \, cosh n). 

*'m 

Likewise 

(82) b m (u) JV^^o m (c X, cos 0) = So m (c*\, cos t;) e^ dv. 



We now differentiate this last relation with respect to and then put 
& = 0. In virtue of (14) the coefficient b m (u) proves to be 



(83) b(u) = Q Boi(coX, cosh u). 



386 CYLINDRICAL WAVES [CHAP. VI 

The complete expansion for a plane wave of arbitraiy direction is, 
therefore, 



(84) e* p = \/87r 2j i m JJM H<4(c X, cosh u) Se m (c X, cos v} 

m-O Um 

e m (coX, cos 0) + -tt Bo^CcoX, cosh u) So m (c \, cos v) So m (c X, cos 0) . 

This last result enables us to write down at once the integral repre- 
sentations of those elementary elliptic wave functions that are finite on 
the axis. We need only multiply (84) by Se m (c<>\, cos ft) and integrate 
over a period. From the orthogonal properties of the functions it follows 
that 

V-m /2* 

(85) KeJ(coX, cosh w) Se m (co\, cos v) = , I /Se m (c X, cos 0) e* p dft; 

VOTT Jo 

similarly, for the odd functions, 

* m /2ir 

(86) JRoi(coX, cosh w) /So w (coX, cos v) = / _ : I So m (c<>\, cos ) 

VSTT Jo 



With the help of (67) it can be shown also that the elementary functions 
of elliptic waves traveling outward from the axis are represented by the 
contour integral 

(87) 

N . /2 f^" , 

BeI(coX, cosh u) $e w (coX, cos v) = z*"-"* * /- I oe OT (c X, cos 8) e iK v dS. 

\Tjo 

The upper sign in the exponential is to be chosen when Tr/2 < v < w/2, 
the negative sign when ir/2 < v < 3w/2. A similar relation can be 
deduced for the odd functions. 

From the analysis of Sec. 6.8 it is easy to see that the even circular 
wave function cos nO /*(Xr), which remains finite on the axis, can be 
represented by the integral 



(88) 27Ti n cosn0J n (Xr) = I cos n/3 e** d0, 

Jo 

where p = x cos j3 + y sin = r cos (/3 0). Upon multiplication by 
DjJ and summation over even or odd values of n } one obtains 



(89) Be^fcoX, cosh u) /Se m (coX, cos v) = */s ^ t""-" 1 !);? cos ndj n (\r) 9 

n 

which expresses the even elliptic wave function as an expansion in circular 
wave functions. A similar expression can be found for the odd functions. 



PROBLEMS 387 

An addition theorem relating circular and elliptic waves referred to 
two parallel axes has been derived by Morse. 1 

Problems 
1. Show that the equation 

ay W 
V-^ -<r M -=0 

is satisfied by wave functions of the type 



^ = Ce i( * l + * J > cos (h& - Si) cos (h z y - Sde**-*, 
h\ + hi + hi - W. 

Let ^ = n, U x n v 0, be the components of a Hertz vector oriented in the direc- 
tion of propagation. Show that this leads to a transverse magnetic field whose 
electric components are 



E x - -tfcifc,C*'< 8l + fa > sin (hix - 50 cos 
.# - -tfc,fc,C f 6^ ai+ **> cos (/lire - Si) sin (fc 
E t - (jfc 2 - fcJ)C6*< ai + 5l > cos (fci - 5i) cos 

and whose impedances are 

E. k* - hi 

Z = - - = O>M cotan 

H y ik*hi 

E, k* -h* 
- 



Zf 



cotan (h 2 y 
ii 

E x o>M^8 



Find the components of a corresponding transverse electric field by choosing 

$ = n*, n* = n* = o. 

The factor e 1 '^ 1 "*" 8 ^ is introduced to include traveling as well as standing waves. 
Thus, if one chooses 5i = too, the field behaves as e lhix . 

2. The equation of a two-dimensional wave motion in a nondissipative medium is 

ay ay i ay _ 

aa? 2 dy 2 a 2 dt 2 
Show that 




)i TT G(x, y\ when - 0. 

at 

This is the Poisson-Parseval formula. Note that at time t the wave function \f/ is 
determined by the initial values g and G at all points on the circumference and the 
interior of a circle of radius at. Thus even in nondissipative media the disturbance 

1 MORSE, loc. tit. 



388 CYLINDRICAL WAVES [CHAP. VI 

persists after the passage of the wave front. This behavior is characteristic of 
wave motion in even-dimensional spaces. 
3. Show that the wave equation 



IL( ^ 

~rdr\ ~dr 



_ 

dz* c 2 dt* 



has a particular solution of the form 



- I F I z 4- ir cos a, t -- sin a ) da, 
2*- Jo \ c ) 



which reduces to ^ = F(z, t) on the axis r = 0, where F is an arbitrary function, 
Show that another solution of the same equation is 



I F [ z r sin/i a, t cosh a J da. (Bateman) 

J->o \ C / 



4. A special case of the solution obtained in Problem 3 is the symmetric, two- 
dimensional wave function. 



I F ( t - - cosh a ) da, 

27T JO \ C / 



which represents a wave expanding about a uniform line source of strength F(t) along 
the z-axis. 

Show that if F(t) = when t < 0, then 



c C ~* F(ft) 

= _ I 

27TJ-C. Vc*tt-W-r* 



which is zero everywhere as long as t < r/c. 

If the source acts only for a finite time T, so that F(t) =0 except in the interval 

< t < T, show that the wave leaves a residue or "tail" whose form, when t ^> T, 

c 

is determined by 




(Lamb, "Hydrodynamics," 5th ed., p. 279, 1924. Cambridge University Press.) 
5. Show that the equation 



-~ + ~~ + - 

is satisfied by 



f 

JH 



+ a) 



r4- ro v P (* x*Y (y i/o) 1 

where 

r* x 2 4- y 2 , rj a?5 + yj. (Bateman) 

6. Show that the equation 



PROBLEMS 389 

is satisfied by 



where r 2 a 2 4 y 2 and / and F are arbitrary functions. To what limitations is this 
solution subjected? 

Obtain in this way the particular solution 

1 

^ = - e~ Xr (x 4* iy) (Bateman) 

r 

7. The equation 

\dx 2 dy 2 / dt dt* 

is reduced by the substitution 

t = e-**u(x, y, 
to 



The initial conditions are 



u = flf(x, y), = G(x, y\ when i = 0. 



Let r be the radius of a sphere drawn about the point of observation and a, /9, 7 the 
direction cosines of r with respect to the x, y, z axes. Show that a solution is repre- 
sented by 

^ /^ 

u(x, y, t) = ~- - I I ig(x + ata, y + atft) cos (ibty) dtt 
ir M J J 



I J 



* GO + ate, y + a*/3) cos (ibty) do, 



(Boussinesq) 

where dn is the solid angle subtended by an element of surface on the sphere. 
8. Show that 



+ i) Jo 



c s (p c s 0) si 



is an integral representation of the Bessel function of the first kind valid for all values 
of the order p provided Re(p) > J. 
9. Derive the integral representations 



cos (z cosh a) dot, 1m z > 0. 





*. - -? r 

^r JO 
for the Bessel function of the second kind and zero order, and 

/* 00 

m a) (s) . ip+i ~ I e"^ ooah a cosh pa da, Im z < 0. 

* Jo 



390 CYLINDRICAL WAVES 

10. Show that the modified Bessel equation 

t ^ _i_ ^L _ j_ 2^ _. 
is satisfied by the function 

/, - 

^T-O 
Where 



[CHAP. VI 



m! F (p + 



1 /2\ p "^ 2m 

+ m + 1) \2/ 



/ p (z) - 



when -T < arg z < - 



The most commonly used independent solution is the one denned by 



A complete set of recurrence relations for these functions are given by Watson, 
"Bessel Functions," page 79. 

11. Demonstrate the following integral representations of the modified Bessel 
functions. 



r(i)r( P 



/ \ p C* 
P + t) V|J Jo 



r(i)r(p 



It is assumed that Re 



( p + - J 



> and jarg z\ < -- 



12. Show that for very large values of the argument the asymptotic behavior 
of the modified Bessel functions is given by 



. 



, m) 



., , 

provided - < arg z < 



m-0 



where 



m 



- 3 a ) 



- (2m - 



m! T(p - m 



PROBLEMS 391 

Retaining only the first term, 

/,(*)~J-L<i, Kp (z)c 

\ tiirZ 

13. Prove that if R * \/ r2 + g2 is the distance between two points, the inverse 
distance is represented by 

-- = - I cos PZ K (pr) dp, 
R * Jo 

where K Q (pr) is the modified Bessel function defined in Problem 10. 

14. As a special case of Problem 4, show that the equation 



/ \ 

r Or \ dr / c 2 



dt* 
is satisfied by 

^(r, - K Q (ikr) e iu>t , 

where k = u/c and KQ is the modified Bessel function defined in Problem 10. This 
solution plays the same role in two-dimensional wave propagation as does the function 
e lkR /R in three dimensions. 
15. Show that the equation 



rdr\ dr 
is satisfied by 

1 . "^1 *?? , , N m 

^ = - Jo(\r) + 2 "^ e n * Jmfrr) cos 0, 
n -^* ~ n 

m=*l n 

which has a period of 2mr in 6 and hence is multivalued about the axis. Show that 
this solution can be represented by the integral 



following a properly chosen contour C in the complex a-plane. This solution is 
everywhere finite and continuous. ^ behaves like a plane wave at infinity provided 
| | < ir, but vanishes (is regular) at infinity if 6 lies between any pair of the limits 

TT < < STT, STT < < 5V, -Zir < < -TT, 

which define the 2nd, 3rd, . . . nth sheets of a Riemann surface. Such multiform 
wave functions have been applied by Sommerfeld to diffraction problems. 

16. Derive explicit expressions for the radial and tangential impedances of elliptic 
cylindrical electromagnetic waves. 

17. Show that the integral 




o 
is a cylindrical wave function which is equal to e ikR /R, R* = r 2 + z*> when /(X) 1. 



CHAPTER VII 
SPHERICAL WAVES 

In the preceding chapter it was shown how the analytic difficulties 
involved in the treatment of vector differential equations in curvilinear 
coordinates might be overcome in cylindrical systems by a resolution of 
the field into two partial fields, each derivable from a purely scalar func- 
tion satisfying the wave equation. Fortunately this method is applicable 
also to spherical coordinates to which we now give our attention. The 
peculiar advantages of cylindrical and spherical systems are a conse- 
quence of the very simple character of their geometrical properties. A 
deeper insight into the nature of the problem and of the difficulties 
offered by curvilinear coordinates in general will be gained from a brief 
preliminary study of the vector wave equation. 

THE VECTOR WAVE EQUATION 

7.1. A Fundamental Set of Solutions. Within any closed domain 
of a homogeneous, isotropic medium from which sources have been 
excluded, all vectors characterizing the electromagnetic field the field 
vectors E, B, D, and H, the vector potential and the Hertzian vectors- 
satisfy one and the same differential equation. If C denotes any such 
vector, then 

(D VC-^-/^-0. 

Because of the linearity of this wave equation, fields of arbitrary time 
variation can be constructed from harmonic solutions and there is no 
loss of generality in the assumption that the vector C contains the time 
only as a factor e~ w *. By the operator V 2 acting on a vector one must 
understand T 2 C = W C V X V X C ; therefore, in place of (1) 
we shall write 

(2) VV C - V X V X C + /c 2 C = 0, 

where fc 2 = /*w 2 + io>cco as usual. 

Now the vector equation (2) can always be replaced by a simultaneous 
system of three scalar equations, but the solution of this system for any 
component of C is in most cases impractical. [C/. (85), page 50.] It 
is only when C is resolved into its rectangular components that three 

392 



SBC. 7.1] A FUNDAMENTAL SET OF SOLUTIONS 393 

independent equations are obtained and in this case 

(3) V 2 C,- + fc'C,- = 0, (j = x, y, z). 

The operator V 2 can then be expressed in curvilinear as well as rectangular 
coordinates. Very little attention has been paid to the determination of 
independent vector solutions of (2), but the problem has been attacked 
recently by Hansen 1 in a series of interesting papers dealing with the 
radiation from antennas. 

Let the scalar function ^ be a solution of the equation 

(4) W + *V = 0, 

and let a be any constant vector of unit length. We now construct three 
independent vector solutions of (2) as follows: 

(5) L = V^, M = V X a*, N = ~ V X M. 

K 

If C is placed equal to L, M, or N, one will verify that (2) is indeed satisfied 
identically by (5) subject to (4). Since a is a constant vector, it is clear 
that M can be written also as 

(6) M = L X a = ~ V x N. 

k 

For one and the same generating function ^ the vector M is perpendicular 
to the vector L, or 

(7) L M = 0. 

The vector functions L, M, and N have certain notable properties 
that follow directly from their definitions. Thus 



(8) v x L = 0, V - L = 
whereas M and N are solenoidal. 

(9) V M = 0, V N = 0. 

The particular solutions of (4) which are finite, continuous, and single- 
valued in a given domain form a discrete set. For the moment we shall 
denote any one of these solutions by \l/ n . Associated with each character- 
istic function \p n are three vector solutions L n , M n , N n of (2), no two of 
which are colinear. Presumably any arbitrary wave function can be 
represented as a linear combination of the characteristic vector functions; 
since the L n , M n , N n possess certain orthogonal properties which we shall 
demonstrate in due course, the coefficients of the expansion can be deter- 

1 HANSEN, Phys. Rev., 47, 13&-143, January, 1935; Physics, 7, 460-465, December, 
1936; /. Applied Phys. t 8, 284-286, April, 1937. 



394 SPHERICAL WAVES [CHAP. VII 

mined. When the given function is purely solenoidal, the expansion is 
made in terms of M and N n alone. If, however, the divergence of the 
function does not vanish, terms in L n must be included. 

The vectors M and N are obviously appropriate for the representation 
of the fields E and H, for each is proportional to the curl of the other. 
Thus, if the time enters only as a factor er*** and if the free-charge density 
is everywhere zero in a homogeneous, isotropic medium of conductivity 
tr, we have 



(10) E = VXH, H = - V X E. 

K* IUJJ, 

Suppose then that the vector potential can be represented by an expansion 
in characteristic vector functions of the form 

(H) A = - ^ (a*M n + &nN n + c n L n ), 

CO dfri i 

n 

the coefficients a n , fe n , c n to be determined from the current distribution. 
By (8), (10), and the relation M H = V X A, we find for the fields 

> H = ~ 



The scalar potential <j> plays no part in the calculation, but can, of course, 
be determined directly from (11). For by (27), page 27, V A == ~ & 2 0; 

CO 

hence, by (8) 



Then V<f> = ~^c n L n and clearly the relation E = V< --- leads 



again to (12). Finally, if we recall that under the specified conditions 
an electromagnetic field can be represented in terms of two Hertz vectors 
by the equations 

(14) E = vxvxn + ico/iV x n*, H = ~ v<x n + v x v x n* 

f 



it is immediately apparent that 

n = 4 2 & *" a > n * = -L 2 a ^ a > 



where a is a constant vector. 

Before applying these results to cylindrical and spherical systems, let 
us consider the elementary example of waves in rectangular coordinates. 



SEC. 7.2] APPLICATION TO CYLINDRICAL COORDINATES 395 

A plane wave whose propagation vector is k = fcn is represented by 

(16) I = 6*-*-** 

where R is the radius vector drawn from a fixed origin. Then since 
k R = kxX + k y y + fc*z, it is easy to see that 

(17) L = tyk, M = tYk X a, N = - ^(k X a) X k. 

In this particular case L M = M N = N L = 0; all three vectors are 
mutually perpendicular and L is a purely longitudinal wave. Now the 
polar angles determining the direction of k are a and /3, as defined in 
Fig. 66, page 362, and general solutions of (2) can be constructed by 
integrating these plane-wave functions over all possible directions. If 
g(aj ff) is a scalar amplitude or weighting factor, one may write for L 
in any coordinate system, subject to convergence requirements, 

(18) L = ie-i** j da ( dp g(a, 0)k(a, p) e*- *; 
likewise for M and N 

(19) M = ie-** I dot \ d&g(a, 0)k X ae* ;R , 

(20) N = ~ e-** (da (dp g(a, j8)(k X a) X k e*' R . 

7.2. Application to Cylindrical Coordinates. 1 The scalar character- 
istic functions of the wave equation in cylindrical coordinates were set 
down in Eq. (6), page 356. There are certain disadvantages in the use of 
complex angle functions exp(in0) and it will prove simpler to deal with 
the two real functions cos nO and sin nO which may be denoted simply 
as even and odd. Cylindrical wave functions, constructed with Bessel 
functions of the first kind and hence finite on the axis, will be denoted by 
^ 1} , while the wave functions formed with N n (\r) or H*(\r) will be 
called i/^ 2) and r/^ 3) respectively. Thus 

= cos nO J n (Xr) e^~-<, 
= sin nO J n (Xr) e**-^, 

= cos 
= sin 

where X = \/fc 2 h 2 = k sin a as ustial. Functions of the first kind are 
1 Compare the results of this section with Eqs. (36) and (37), p. 361. 



396 SPHERICAL WAVES [CHAP. VII 

represented by the definite integrals 

i-n C 2 * 

= 1- e * r cos <*-*> cos up dp e ih *-** f 
Z " J Q 

7 --n C 2 * 

= 7T- e * r cos ( 

rt Jo 



sn 



and representations of other kinds differ from these only in the choice 
of the contour of integration. 

From (23) we now construct integral representations of the vector 
wave functions. Thus 



Upon differentiating (23) with respect to r, 0, and z and noting that 
k sin a cos (0 6), k sin a sin (0 - 0), and fc cos a are the components 
of k respectively along the axes defined by the unit vectors ii, i 2 , is, so that 

(25) k = iifc sin a cos (0 - 0) + i 2 fc sin a sin (0 - 0) + i 3 & cos a, 
we find for the even function 



i\-n C 2 " 

L x = ~ 

ZTT Jo 



(26) L x = ~ ke tV cos tf~'> cos 

ZTT Jo 

which is of the form (18). The corresponding odd function is obtained 
by replacing cos up by sin n/? under the integral. When applying these 
characteristic functions to the expansion of an arbitrary vector wave, 
it proves convenient to split off the factor exp(*7i2 io>0, and we shall 
define the vector counterparts of the scalar function . /(r, 0) of Chap. VI by 



(27) L n = l n e^-^, M n = m n e ihg -^ 9 N n = n n e^~K 
From (26) we have 

VI n /*2w nnct 

(28) li = V- fa*- cos <"-> ^ n/5 dj8. 

o nX ^ Jo S111 

In like fashion one finds integral representations of the independent 
solutions. For the constant vector a we choose the unit vector i 3 directed 
along the 2-axis. Then 



(29) M< = W X i, = - -- ii 
whence it is easily deduced that 

VI n /*27r 

(30) m> = ^ ( (k X We*' - - 



SEC. 7.2] APPLICATION TO CYLINDRICAL COORDINATES 397 

since V X M n = &N n , we have 

(31) n<* = pr ( 2 * (Art, - Ak)6> ^s 0-0 cos 

x 27T/C J sin K 

From these integrals one can easily calculate the rectangular components 
of the wave functions. The vector functions of angle in the integrands 
are resolved into rectangular components which can then be combined 
with cos n/3 or sin nf$. The resulting scalar integrals for the components 
are evaluated by comparison with (23). 

To expand an arbitrary vector function of r and 6 in terms of the 
la , m a , n e , one must show that these functions are orthogonal. Let 

o nX o nX o nX 

ii> i2> is be unit base vectors of a circularly cylindrical coordinate system, 
Fig. 7, page 51, and Z n (\r) a cylindrical Bessel function of any kind. 
Then by (5) we obtain 

(32) * A = I: Z W ^ nB ^ + l 2 n (Xr) Sin n ii + ih ^(Xr) c . os nS i 3 , 

n Or oiii T cos sin 

(33) m., - T 5 S.(Xr) ^ n 1, - 1 Z n( Xr) ^ n. i 2 , 

< 34 > V = * I ^M sin ^ ^ T W Z *M S ^ i2 + T Z -^> 

c . os n^ i s . 
sin 

Now it is immediately evident that the scalar product of any two func- 
tions integrated over 6 from to 2?r must vanish if the two differ in the 
index n. 

(*2ir 1* 

(35) lenx U <W = 1. - l e , v dB = 0, if n ^ n'. 

JO JO nX o nX 

Let us understand by 1 the function 1 with the sign of ih reversed. 1 Then 
the normalizing factor is to be found from the integral 



+ Z n (\r) Z n (\'r) + hh'ZM Z n (\'r) 

where 5 = unless n = 0, in which case 5 = 1. The first two terms on 
the right can be combined with the aid of the recurrence relations (24) 
and (25), page 359, giving 

1 This is not necessarily the conjugate, since X and h may be complex quantities. 



398 SPHERICAL WAVES [CHAP. VII 

(37) JJ' l, nX L nX , dO = (1 + 5)* {^ [Zn-i(Xr) ^(X'r) 



The same recurrence relations lead us also to the integrals 

/2ir /2* 

(38) mn\ monx dO = m, m e ,. , dO = 0, (n ^ n') ? 

JO JO o wA o nA 

/*2r \\/ 

(39) m. m. x , cW = (1 + S)TT ^ [*_i(Xr) Zn-i(X'r) 

JO o nA o** * 

+ Z n+1 (Xr) ^i(XV)], 

A2ir /2r 

(40) n en x n,,^ d(? = n . n, de = 0, (n * n'), 

JO JO o" A o"^ 

(41) J^ 2 ' n. nX & >x , d0 = (1 + 5) |^-' [Z n _ 1 (Xr) Z^,(X'r) 

+ Z B+1 (Xr) Z n+ ,(X'r)] + (XX') 2 2n(X7-) Z.(X' 
Furthermore 

(42) r i- v ^ = r "v > ^ - r v v ^ = 

and by use of the recurrence relations once more 

/2r /2x 

(43) 1 -m o x ,dO= \ m. -n. dtf = 0. 

JO nX e nX JO nA e nA 

There remains only one other combination, and this, unfortunately, leads 
to a difficulty. 

(44) P' 1, fi, d^ = f (1 + 5) ~ UX'' Z n (Xr) Z n (X'r) 

JO o o n K \ 

- ~ [Z^i(Xr) Zn-i(XV) + Z n+1 (Xr) Z n+1 (X' 
a quantity not identically zero. The set of vector functions l e . , m c . 

o"* o nX 

n, , therefore, just fail to be completely orthogonal with respect to 6 

o* A 

because of (44). In many cases this is of no importance, for if a vector 
function is divergenceless as are the electromagnetic field vectors in 
the absence of free charge it can be expanded in terms of m f ^ n e x 

o o 

alone. A complete expansion of the vector potential, on the other hand, 
requires inclusion of the 1 6 * l 

o nA 

1 The completeness of the set of vector functions has not been demonstrated but 

presumably follows from the completeness of the scalar set t, . 

IA 



SEC. 7.3] ELEMENTARY SPHERICAL WAVES 399 

In the case of functions of the first kind the Fourier-Bessel theorem 
has been applied by Hansen to complete the orthogonality and to simplify 
the normalization factors. From (53) and (54), page 371, after an 
obvious interchange of variable, we have 

(45) /(X) = f "rdrf Q "\' dX7(XO/n(XV)/ n (Xr). 

This is applied to (37), (39), (41), and (44) when Z n (\r) is replaced by 



/ n (Xr). For /(X') we shall have X', h' = Vk 2 - X /2 , A'X' and the like. 
Now one must note that the validity of the Fourier-Bessel integral has 

been demonstrated only when f |/(X)|\/X d\ exists, and that this con- 

/o 

dition is not fulfilled in the present instance. The artifices employed to 
ensure convergence involve mathematical difficulties beyond the scope 
of the present work. With certain reservations one may introduce an 
exponential convergence factor exactly as in the theory of the Laplace 
transform on page 310. By 1< 1} X let us understand henceforth the limit 

o n 

approached by IQe-W as s - 0. Then 

* 



(46) lim f " r dr f " X' d\' [X'-l'l^/.(\'r)J.(Xr) = X. 

a _o J J 

Consequently the normalizing factors reduce to 

o T '- *y ^ dr de=(1 
m ?x m -:v x ' ^ dr de=:(1 

fo ~fo T n > fi ^' d *r drde=(l 
while for the troublesome (44) we find 1 



THE SCALAR WAVE EQUATION IN SPHERICAL COORDINATES 
7.3. Elementary Spherical Waves. Since an arbitrary time variation 
of the field can be represented by Fourier analysis in terms of harmonic 
components, no essential loss of generality will be incurred hereafter by 
the assumption that 

(1) * = /(, , *)*-*" 

1 The reader must note that the limit approached by (46) as s -* is not necessarily 
equal to the value of the integral at s =0, for this would imply continuity of the func- 
tion defined by the integral in the neighborhood of s = 0. This point underlies the 
whole theory of the Laplace transform. Cf. Carslaw, " Introduction to the Theory 
of Fourier's Series and Integrals," 2d ed., p. 293, Macmillan, 1921. 



400 SPHERICAL WAVES [CHAP. VII 

In a homogeneous, isotropic medium the function f(R, 0, <t>) must satisfy 

(2) v 2 / + fc2/=0, 

which in spherical coordinates is expanded by (95), page 52, to 

w p as ( B> I) + pan B ( 8in ' $ + PISH + ** - " 

The equation is separable, so that upon placing / = /i(r)/ 2 (0)./3(<) one 
finds 

(4) ft 2 ~& + 2R & + (* 2 B J - 7^ 2 )/i = 0, 



The parameters p and q are separation constants whose choice is 
governed by the physical requirement that at any fixed point in space 
the field must be single- valued. If the properties of the medium arc 
independent of equatorial angle <, it is necessary that/ 3 (</>) be a periodic 
function with period 2ir and q is, therefore, restricted to the integers 
m = 0, 1, 2, . . . . To determine p we first identify the solution / 2 
as an associated Legendre function. Upon substitution of r/ = cos 0, 
Eq. (5) transforms to 



an equation characterized by regular singularities at the points rj = 1, 
q = +i j y = oo, and no others. Its solutions are, therefore, hyper- 
geometric functions. Now when m = 0, (7) reduces to the Legendre 
equation. There are, of course, two independent solutions of the 
Legendre equation which may be expanded in ascending power series 
about the origin rj = 0. These series do not, in general, converge for 
V) = 1. If, however, we choose p 2 = n(n + 1), where n = 0, 1, 2, 
. . . , then one of the series breaks off after a finite number of terms 
and has a finite value at the poles. These polynomial solutions satisfying 
the equation 

are known as Legendre polynomials and are designated by P n (i?). If 
now we differentiate (8) m times with respect to 17, we obtain 

(9) (1 - 7? 2 ) - 2(m + l)ij ^ + [n(n + 1) - m(m + l)]w = 0, 
arj dt\ 



SBC. 7.3] ELEMENTARY SPHERICAL WAVES 401 

where w = d^/dy, and upon making a last substitution of the dependent 

m 

variable, w = (1 if) ^(T?), we obtain (7) in the form 

(10) (1 tf) -~| 2j -~ -f n(n + 1) - /2 = 0. 

The solutions of (10) which are finite at the poles 77 = 1 and which are, 
consequently, periodic in are the associated Legendre polynomials 

(11) 



For each pair of integers there exists an independent solution 
which becomes infinite at rj = 1, and which consequently does not 
apply to physical fields in a complete spherical domain. 

The definition of the associated Legendre polynomial as stated in (11) 
holds only when n and m are positive integers. The functions of negative 
index are related in a rather simple fashion to those of positive index, 
but we shall have no need of them here. To obviate any confusion on 
this matter we choose for particular solutions of (6) the real functions 
cos ra<, sin m<, and restrict m and n to the positive integers and zero. 
It is clear from (11) that P"(i?) vanishes when m > n, for P n (ij) is a 
polynomial of nth degree. We have in fact 



The properties of the hypergeometric functions, of which the Legendre 
functions are the best known example, have been explored in great detail 
and it would be scarcely possible to give an adequate account of the 
various series and integral representations within the space at our dis- 
posal. We shall set down only the indispensable recurrence relations, 

(n - m + IJPjn - (2n + l)ijP + (n + m)P^. 1 = 0, 





~ n - m 
+ (n + m) 
m + l)ip - (n - m + DP-.,, 

(n + m)(n - m 

,__^ _ 1 _ pm4-n 

n 2n + l ^ n+1 ^-i/' 

= n[(n - m + l)(n 



402 SPHERICAL WAVES 

To these may be added the differential relations 

(n m + 



[CHAP. VII 



(14) 



= (n 



dp n _ /i s dp % 

~W ~ -Vl- n -^ 



= g n - m n 

It may be remarked in passing that these relations are satisfied also by 
functions of the second kind Q. 

The functions cos m<t> Pjj(cos 0) and sin m<f> P(cos 0) are periodic on 
the surface of a unit sphere and the indices m and n determine the number 
of nodal lines. Thus when m = 0, the field is independent of the equa- 
torial angle <. If n is also zero, the value of the function is everywhere 
constant on the surface of the sphere. When n = 1, there is a single 
nodal line at the equator = ?r/2 along which the function is zero. When 






FIG. 69. Nodes of the function sin 3?> P*(cos 0) on the developed surface of a sphere. 
The function has negative values over the shaded areas. 

n = 2, there are two nodal lines following the parallels of latitude at 
approximately 6 = 55 deg. and = 125 deg., so that the sphere is 
divided into three zones; the function is positive in the polar zones and 
negative over the equatorial zone. As we continue in this way, it is 
apparent that there are n nodal lines and n + 1 zones within which the 
function is alternately positive and negative. For this reason the 
P n (cos 0) are often called zonal harmonics. If now m has a value other 
than zero, it will be observed upon examination of Appendix IV that 

TO 

the function is zero at the poles due to the factor (1 i? 2 ) 2 , and that the 
number of nodal lines parallel to the equator is n m. Moreover, the 
function vanishes along lines of longitude determined by the roots of 
cos m<t> and sin m<t>. There are obviously m longitudinal nodes that 
intersect the nodal parallels of latitude orthogonally, thus dividing the 



SEC. 7.3] ELEMENTARY SPHERICAL WAVES 403 

surface of the sphere into rectangular domains, or tesserae, within which 
the function is alternately positive and negative. For this reason the 
functions cos m<t> P(COS 0) and sin m<t> P(cos 0) are sometimes called 
tesseral harmonics of nth degree and rath order. There are obviously 
2n + 1 tesseral harmonics of nth degree. This division into positive and 
negative domains is illustrated graphically in Fig. 69 for the function 
sin 3</> Pf (cos 0). 

If the tesseral harmonics are multiplied by a set of arbitrary constants 
and summed, one obtains the spherical surface harmonics of degree n 
with which we have already had something to do in Chap. III. 

n 

(15) 7n(0, 0) = 5) (nm COS m(t> + b nm SHI W0)P(COS 0). 

w = 

The tesseral harmonics form a complete system of orthogonal func- 
tions on the surface of a sphere. 1 It is in fact easy to show through an 
integration of (12) by parts that 



= o, 

when n 7^ I or m ^ I respectively, and that 
f 

J-i 

P - 

J-i L nWJ 1 - 7? 2 m(n - 



From these relations follows the fundamental theorem on the expansion of 
an arbitrary function in spherical surface harmonics: Let g(0, $) be an 
arbitrary function on the surface of a sphere which together with all its first 
and second derivatives is continuous. Then gr(0, 0) oan be represented by 
an absolutely convergent series of surface harmonics , 



(18) 0(0, *) = 5) [a no P n (cos 0) + ] (a nm cos w</> + b nm sin m<) 

n0 wi = l 

P-(cos 
whose coefficients are determined by 

2n + X f f gr(0, 0) P n (cos 0) sin d0 d^, 

?r Jo JO 



a no = 



(19) a n 



m) 



! f f' 

l Jo Jo 



1 The proof of this statement and of the expansion theorem which follows will be 
found in Courant-Hilbert, "Methoden der mathematischen Physik," 2d ed., Chap. 



404 SPHERICAL WAVES [CHAP. VII 

In the case of a function that depends on alone, the conditions deter- 
mining the convergence of an expansion in Legendre polynomials are 
identical with those governing the convergence of a Fourier series. 
Under such circumstances it is sufficient that g(B) and its first derivative 
be piecewise continuous in the interval ^ 6 < 2w. The convergence 
theory of the expansion (18) in surface harmonics, on the other hand, 
presents definitely greater difficulties and the extension of the theorem to 
discontinuous functions involves considerations beyond the scope of the 
present outline. 

There remains the identification of the radial f unction fi(R) satisfying 
(4). If for /i we write /i = (kR)~*v(R), it is readily shown that v(R) 
satisfies 



and hence, by Sec. 6.5, is a cylinder function of half order. 
(21) /i(B) 



The characteristic, or elementary, wave functions which at all points on 
the surface of a sphere are finite and single-valued are, therefore, 

(22) /. mn = -J Z n +*(kR) P?(cos 9) m*. 



As in the cylindrical case, we choose for Z n +i(kR) a Bessel function of the 
first kind within domains which include the origin, a function of the third 
kind wherever the field is to be represented as a traveling wave. 

7.4. Properties of the Radial Functions. Various notations have been 
employed at one time or another to designate the radial functions 
(kR)~*Z n +i(kR) but none appears to have gained general acceptance. 1 
What seems to be a logical proposal has been made recently by Morse 2 
and will be adopted here. Accordingly we define the spherical Bessel 
functions by 



jn(p) = 

(23) n n (p) = 



VII, 6, 1931. For greater detail see Hobson, "The Theory of Spherical and Ellip. 
Boidal Harmonics," Cambridge University Press, 1931. 

^See WATSON, "Bessel Functions," p. 55, Cambridge University Press, 1922. 

3 MOBSB: "Vibration and Sound," p. 246, McGraw-Hill, 1936. 



SBC. 7.4J PROPERTIES OF THE RADIAL FUNCTIONS 405 

Series expansions oi these functions about the point p = can 
be obtained directly from (8) and (11), page 357. If we recall that 
r(2 + 1) SB zT(z), T() = V^, and make use of the duplication formula 



we find for j n (p) 



whence it is apparent that j n (p) is an integral function. Likewise from 
the relation N n +(p) = ( l) n ~ 1 J r . n _ i (p), we obtain for the function of 
the second kind 

((>( ^ /x 1 ^sh r(2n 

(26) n n (p) = -_ 



2V*- 1 J rn\T(n - m + 1) p ' 

We turn next to representations when p is very large, and discover at 
once a notable property of the Bessel functions whose order is half an 
odd integer. On page 358 expansions in descending powers of p were 
given for the functions of arbitrary order p. Such series satisfy the 
Bessel equation formally, but are only semiconvergent. We observe 
now, however, that when p = n + 1/2, the series P n +*(p) and Q n +$(p) 
break off, so that there is no longer question of convergence. Conse- 
quently (13) and (14) of page 358 are analytic representations of /-H(P) 
and N n+ $(p) ; furthermore, it is apparent that these functions of half order 
can be expressed in finite terms. From (23) above and Eqs. (13) to (17) 
of page 358, we obtain 

(27) j n (p) = 1 

(28) rc n (p) = 1 



cos p - - Q n+i(p) sin p _ 

in (p - 11 *) + Q B+J (p) cos 



P n +i(p) sin 
P 

where 
(29) 

n(n 2 - l)(n 2 - 4)(n 2 - 9)(n + 4) 

(*n\ n i \ ( + !) n(n z - l)(n 2 - 4)(n + 3) . 
(30) Q n+} (p) = -Tr-TTi- 2-3lp + ' ' 




For the functions of the third and the fourth kind this leads to 
(31) 



406 SPHERICAL WAVES [CHAP. VII 

These series converge very rapidly so that for large values of p the first 
term alone gives the approximate value of the function. Thus, when 



- / N , x - 

j n (p) ~ - cos ( p - ^ TT 1; ttn(p) ^ - sm ( p 



P P 

The recurrence relations satisfied by the spherical Bessel function 
follow directly from Eqs. (24) to (27) of page 359. 

(33) Zn-l + Zn+l = -^ - Zn, 

(34) j- z n (p) = 2n + l [ nZn ~ l - ( n + IK+i], 

(35) ^ [p" +1 Zn(p)] = P^^n-l, ^ [p~ n Z n (p)] = -p-Z n+1 . 

Having defined the radial functions, we can at last write down the 
general solutions of (3) as sums of elementary spherical wave functions. 
In case /(ft, 0, <) is to be finite at the origin, we have 



(36) / (1) (#, 0, 4>) = ] J(*fl) noPn(cos 0) + (o w cos 



nm sn 

while a field whose surfaces of constant phase travel outward is repre- 
sented by / (3) CK, 0, 0), obtained by replacing j n (kK) by h(kK) in (36). 
A spherically symmetric solution results when all coefficients except a o 
are zero. Then apart from an arbitrary factor, we have 

/(1) _ sin kR f(2) __ cos kR 
Jo ~ kR ' ; ~ kR ' 
^ ' I 1 

/(3) _ x pikR f(4) _ x p -ikR 

/0 ~ kR e > /0 ~kR e ' 

7.5. Addition Theorem for the Legendre Polynomials. If g(0, <) is 
any function satisfying the conditions of the expansion theorem (18), 
its value at the pole, = 0, must be 

(38) \g(0, 4>)] M = 2 an = ^2 (2n + 1} J 

sin d0 d</>, 

since P n (l) = 1, P (1) == 0. In particular let us take for 0(0, </>) any 
surface harmonic F n (0, <) of degree n expressed by (15). Then in 



SEC. 7.5] 



ADDITION THEOREM 



407 



virtue of the orthogonality relations, the sum in (38) reduces to a single 
term and we obtain the formula 



(39) 






F n (0, 4>)P n (cos 0) sin 6 dO d<t> = 






We shall apply this result to the problem of expressing a zonal har- 
monic in terms of a new axis of ref- 
erence. Let P in Fig. 70 be a point 
on a sphere whose coordinates with 
respect to a fixed rectangular refer- 
ence system are and <. A second 
point Q has the coordinates a and 
0. The angle made by the axis OP 
with the axis OQ is 7. The zonal 
harmonics at P referred to the new 
polar axis OQ are of the form P n 
(cos 7) and our problem, therefore, 
is to expand P n (cos 7) in terms 
of the coordinates 6, <, and a, 0. 

It is apparent that on a unit 
sphere cos y is the projection of the 
line OP on the axis OQ. If #, y, z 
are the coordinates of P, #', t/', z' those of Q, then 

(40) cos y = xx' + yy r + zz r = sin 8 sin a cos (< /S) + cos cos a. 
We now assume for P n (cos 7) an expansion of the form 




FIG. 70. Rotation of the axis of 
"om OP to OQ for a system of 



(41) P n (cos 7) = r Pn(cos 



cos 



sn 



multiply both sides by P(COS 6) cos w</>, and integrate over the unit 
sphere. With the help once more of the orthogonality relations, we 
obtain 



(42) I I P n (cos 7) P?(cos 0) cos m<t> sin 6 dd d<}> 
Jo Jo 



27T 



But by (39) 



(43) 



f 2 * r 

I I P n 

Jo Jo 



(cos 7)P(cos 0) cos m< sin dS d$ 



47T 



= 2n + l t P ( cos fl ) cos w *k-o = 2n + 1 jP ^( cos a ) cos m ^ 



408 SPHERICAL WAVES [CHAP. VII 

since when 7 = 0, 6 = a, <f> = ft. Consequently, 



(44) c = 2 - P?(cos ) cos m0. 
Likewise, 

(45) <Z m = 2 fr ""*?] P?(cos a) sin m/3. 

^/t "T* 171) I 

The desired expansion, or addition formula, is therefore 

(46) P n (cOS 7) =* Pn((X>S ) P n (cOS 0) 



s a > p (cos *> cos m( * ~ "5. 

V** I "V 

m * 1 

This result leads to an alternative formulation of the expansion 
theorem stated on page 403. If g(0, <) and its derivatives possess the 
necessary continuity on the surface of a sphere, its Laplace series (18) is 



(47) g(0, 4>) = 
or by (19) and (46), 

1 ^ri C 2 * C* 

(48) g(6, <t>) = -=- >j (2n + 1) g(a, ^3)Pn(cos 7) sin a da d)8. 

tfTo ^ ^ 

7.6. Expansion of Plane Waves. It is now a relatively easy matter 
to find a representation of a plane wave propagated in an arbitrary 
direction in terms of elementary spherical waves about a fixed center. 
The direction and character of the plane wave will be determined by its 
propagation vector k whose rectangular components are 

(49) fci = fc sin a cos 0, fc 2 = k sin a sin jS, fc 3 = k cos a, 
while the coordinates of any arbitrary point of observation are 

(50) x = R sin 6 cos 0, t/ = R sin sin <, z = R cos 0. 
Then the phase is given by 

(51) k R = fcft[sin a sin cos (* 0) + cos a cos 0] = fc/J cos 7. 

The f unction / (l) (K, 0, </>) = exp(tfc J R cos 7) is continuous and has continu- 
ous derivatives everywhere including the origin R = 0. It can, therefore, 
be expanded according to (36), page 406. Consider first an axis that 
coincides with the direction of the wave. Since the wave is symmetrical 
about this axis, we write 



(52) giMeo., = an j B (fcfl)P n (cos 7). 

n-0 



SBC. 7.7] INTEGRAL REPRESENTATIONS 409 

The coefficients are determined in the usual manner by multiplying 
both sides by P n (cos 7) sin 7 and integrating with respect to 7 from to 
TT. . Then by (17) 

2n 4- 1 C* 
(53) a n j n (kR) = f- J Q e< * y P n (cos 7) sin 7 dy. 

To rid this relation of its dependence on R, differentiate both sides with 
respect to p = kR and then place p = 0. From (25) it follows that 



o ~ (2n+l)P 
hence, 

in Jo C sn y P " 



(2n + 1)1 " 



sn 



The integral on the right of (55) is readily evaluated and we obtain 
a n = (2n + l)t n > or 



(56) e** C08 * = 2J <(2n + l)j n (kR) P n (cos 7). 

n-O 

In case the z-axis of the coordinate system fails to coincide with the 
direction of the plane wave, one may use the addition theorem (46) to 
express (56) in terms of an arbitrary axis of reference. Then 

(57) e* R cos Y = 2 i"(2n + l)jn(*B) [P n (cos a) P n (cos 0) 

n-O L 

n 

_j_ 2 ^. -7 T; riP(cos a ) Pn( cos ^) cos m(<t> ft) 

m-1 ^ ^ ' J 

7.7. Integral Representations. We have found it convenient on 
various occasions to represent a wave function as a sum of plane waves 
with appropriate weighting or amplitude factors. The direction of each 
component wave is determined by the angles a and ft. Integration is 
extended over all real directions in space, and may in certain cases include 
imaginary values of a and ft as well. As in Sec. 6.7, we wish to find 
representations of the form 



(58) f(x, y, z) = f da C dft g(a, ft) 

or, when x = R sin 6 cos <, y = R sin sin <, z = R cos 0, 

(59) /(, 0, 0) = / da f dft g(a, ft) e** *, 

where a and ft are the angles shown in both Figs. 66 and 70. In the 
case of cylindrical waves a fixed frequency and a fixed wave length along 



410 SPHERICAL WAVES [CHAP. VII 

the z-axis lead to a constant value of a, so that the directions of the plane 
waves representing a cylindrical function f(u l 9 u 2 ) constitute a circular 
cone. Since for spherical waves there exists no such preferred' axis, the 
integration must be extended to both a and ft. 

An integral representation of elementary spherical wave functions 
can be obtained directly from the expansion (57). If both sides of that 
equation are multiplied by P(COS a) cos mfi or by P(COS a) sin ra/3, one 
obtains, thanks to the orthogonality relations (19), the result 



(60) j n (*fi) P?(cos 0) n S m<t> = 

sin a da. dfl. 

Upon multiplying both sides by the arbitrary constants a nl , &nm and 
summing over m, this can be written also as 

/*2ff /*7T 

(61) j n (kR)Y n (6, *) = p| e* B coa ^F n (, 0) sin a da dj8. 

* ** t/ *J 

From these general formulas may be derived a number of useful special 
cases. Thus by choosing m = 0, 6 = 0, we obtain a representation of the 
spherical Bessel function j n (kR). 

i~ n C* 

(62) j n (kK) = -TT c* 5cOBa P n (cos a) sin a da, 

^ Jo 

or 
{63) 

It is in fact easy to show that Eq. (4), page 400, is satisfied by the integral 

(64) z n (kR) = t- f e**'P w (iy) <^, 

Jc 

provided the contour C is such that the bilinear concomitant 



(65) 



= 
c 



vanishes at the limits. In place of 77 = 1, we may choose a value that 
will cause the exponential to vanish. Thus if k is real, we take for the 
upper limit rj = i and obtain the representations 



= f- e'^P n (7,) dr,, 

(66) J r. 

= i-" 

/"""* 



SEC. 7.8] A FOURIER-BESSEL INTEGRAL 411 

If on the other hand k is complex due to a conductivity of the medium, 
one may choose that root of A; 2 whose imaginary part is positive and 
replace the limit 17 = i oo by t\ = oo . 

Again from (60) and (54) , placing < and R equal to zero, one obtains 
a representation of the function P(cos 6) : 

(67) P(COS 0) = n ' | 2 I I cos n 7 P(COS a) cos mft sin a da dp. 

The integration with respect to a cannot be carried out easily with only 
the help of formulas from the present section at our disposal, but by other 
means it may be shown that (67) reduces to 

(68) P?(cos 0) = ^^ i~ m I (cos 6 + i sin 6 cos j8) cos mp dp. 



Finally, one will note that by making use of the integral representa- 
tion (37), page 367, for a Bessel function, the right-hand side of (60) can 
be reduced to a single integral. When <f> = 0, 

i-n C* 

(69) jn(fcjR) PjT(cos 0) = I e ** c08 cos J m (kR sin d sin a) 

^ Jo 

a) sin a da. 



7.8. A Fourier-Bessel Integral. It will be assumed that the arbitrary 
function f(x, y, z) and its first derivatives are piecewise continuous and 
that the integral of the absolute value of the function extended over all 
space exists. Subject to these conditions the Fourier integral of /(#, ?/, z) 
is 



(70) f(x, y, z) = _ w _ g(k lt k,, 

dki dk 2 
and its transform is 

(71) g(k l , ft,, /c,) = (I) 1 J_"_ J_" a J_^ /(x, y, )-.-(*+**+.) 

dx dy dz. 

We now transform these integrals to spherical coordinates under the 
tacit assumption that a triple integral extended over an infinite cube can 
be replaced by an integral over a sphere of infinite radius. Proceeding 
as in Sec. 6.9 and noting that k\x + k^y + k& = kR cos 7, we obtain 



(72) /(, 0, <) = Q-Y J^ f ' f 2 ' g( a9 p t k) e<* - y fc 2 sin a dk da dp, 

(73) g( a , P, k) = Qj J J j /(ft, 0, <) e- C09 > 72 2 sin 6 dR de d*. 



412 SPHERICAL WAVES [CHAP. VII 

Next we shall assume that f(R, 0, <t>) =*f(R)Y n (0, tf>) and impose 
upon the otherwise arbitrary function f n (R) the condition that it shall be 
piecewise continuous and have a piecewise continuous first derivative, 

and that the integral f |/(B)| dR shall exist. Then 



(74) g(a, ft ft) = Q J2 2 dR f n (R) Y n (6, *)- 

sin d6 d<, 

which in virtue of (61) and the fact that j n (kR) = ( l) n j n (kR) goes 
}ver into 

75) 0(, ft fc) = (- l)7n(a, 0) " /n(B) jn(fcB)B dR 



or gf(a, j8, ft) = ( I) n 7 n (a, ft) g n (k). Introducing this evaluation of the 
transform back into (72) and again interchanging the order of integration, 
we are led to 



(76) f n (R) 7 n (fl, *) = g n (k)j n (kR)k*dk 

\' Jo 

and hence to the symmetrical pair of transforms 

(77) /() = Jfe f g(fc) jn(ftK)fc 2 dfc, 

X 71 " Jo 

(78) g(k) = 



from which the subscript n has been dropped as having no longer any 
particular significance. In the special case n = ; Eqs. (77) and (78) 
reduce to the ordinary Fourier integral (18), page 288. In case the 
chosen value of R coincides with a point of discontinuity mf(R), we write 

(79) \ f k*dk f /( P ) j n (fcp) j n (kR) p 2 dp = I (f(R + 0) + f(R - 0)]. 
^Jo Jo ^ 

7.9. Expansion of a Cylindrical Wave Function. When calculating 
the radiation from oscillating current distributions it is sometimes 
necessary to express a cylindrical wave function in spherical coordinates. 
The integral representation of wave functions that are finite on the axis is 



(80) m<f> J.(Xr) *** = 
Now 

(81) k R = Xr cos (ft 4>) + ifez = fcfi[sin a sin cos (0 0) 

+ cos a cos 0], 



SEC. 7.10] ADDITION THEOREM FOR z<*(kR) 413 

and in (57), page 409, we have an expansion in the appropriate spherical 
wave functions of exp(zk R). Upon integration with respect to 0, we 
obtain 

(82) 



, 

s a) P"(cos 0) j n (kR). 



Since P vanishes if m > n the first m terms of (82) are zero and the 
expansion can be written 

00 

fQ1\ T f\v\ nih* ^^ f 'n/n^ i rk__ i i\ ^ 

(OO) J m \W) 6 * ^>j f 

n=*0 

P^(cOS0)j n +,n(2). 

When a = w/2, h = 0, X = k, and 

m (o\ = 0> if n is odd ' 

n+wWy ^ 4- 9tn 1M 

if n is even. 

2 n+m-l(!M ! (" 



The result is an expansion of a cylindrical Bessel function in a series of 
spherical Bessel functions. 

J flfert 
./.(AT; 



7.10. Addition Theorem for z (kR). Let P(B , 0, *) be a point of 
observation and Q(Ri, Oi, <t>i) the source point of a spherical wave. The 
coordinates J? , ^, and Ri, Oi, <t>i are referred to a fixed coordinate system 
whose origin is at 0. The distance from Q to P is, therefore, 



^2 = : *v jRg ~h ^2f 2/?o^?i cos 7 

where cos 7 = cos cos 0i + sin sin 0i cos (<#> <#>i). Our problem is 
to express a wave function referred to Q as a sum of spherical waves whose 
source is located at 0. General treatment of this problem demands a 
lengthy calculation and its practical importance is insufficient to warrant 
discussion here. However, in the theory of radiation we shall have occa- 
sion to apply the special case in which the wave is spherically sym- 
metric about the source Q. One finds then without great difficulty the 
expansions: 

O1V\ 1 /? * >^ 

l)P n (cos 7) j* 

ff^O 



414 SPHERICAL WAVES [CHAP. VII 

s>ikR 



(87) 

00 

(2n + l)P(cos 7) J 



n-O 

(2n + l)P n (cos 7) j(*Bi) WB ), (Bo > Bi). 

The proof is left to the reader. 

THE VECTOR WAVE EQUATION IN SPHERICAL COORDINATES 

7.11. Spherical Vector Wave Functions. As in the case of cylindrical 
coordinates, one can deduce solutions of the vector wave equation in 
spherical coordinates directly from the characteristic functions of the 
corresponding scalar equation. Following the notation of the preceding 
section we shall put $ e = /, e"*"', where f e is the characteristic 

Q mn O mn Q mn 

solution 

(1) /. = ? Q S m<P;r(cos0)z n (fctf). 

o mn 

According to Sec. 7.1, one solution of the vector wave equation 

(2) VV C - V X V X C + /c 2 C = 

can be found simply by taking the gradient of (1). We define L = V^, 
and split off the time factor by writing L = le-**'. Then by (95), page 
52, 



(3) l. mn = ^g Zn(fctf) P?(cos 0) n 






where ii, i 2 , is are the unit vectors defined in Fig. 8, page 52, for a 
spherical coordinate system. 

To obtain the independent solutions M and N as described in Sec. 7.1, 
one should introduce a fixed vector a. Such a procedure in the present 
instance is of course permissible, but then M and N will be neither normal 
nor purely tangential over the entire surface of the sphere. If in place 
of a we use a radial vector ii, a vector function L X ii is obtained which is 
tangent to the sphere; but ii is not a constant vector and consequently it 
by no means follows & priori that it can be employed to generate an 
independent solution. However, we shall discover that a tangential solu- 
tion M can in fact be constructed from a radial vector. Unfortunately 
the procedure fails in more general coordinate systems. 



SEC. 7.11] SPHERICAL VECTOR WAVE FUNCTIONS 415 

Let us try to find a solution of (2) in the form of a vector 
M = V X (iiu(BM = L X iiu(R), 

where u(R) is an unknown scalar function of R. Then if Mij M z , M* 
are the R, 0, and < components of M, we have 



The divergence of M is zero and we now expand the equation 
V X V X M - fc 2 M = into 72, 6, and </> components by (85), page 50. 
The ^-component is identically zero, whatever u(R). The conditions 
on the and <j> components are satisfied provided u(R) is such that 

/K\ d * ( t\ . l d \ ' a d f IN! 1 d 2 / t\ 

(5) C^+j^^^smfl^ 



= 0. 

(The peculiar advantage of spherical coordinates over more general 
systems is that the conditions on both tangential components are iden- 
tical.) If, therefore, we choose u(R) = R, Eq. (5) reduces to the required 
relation 

(6) VV + fcV = 0. 

Hence in spherical coordinates, (2) is satisfied by 

(7) M-VXRiA = LxR = ivxN, 
whose components are 

- 



From the relation fcN = V X M the components of a third solution can 
be easily found. 

/nN ... aw) .,.,,. Ar i 3 2 (W Ar i a*W) 

(9) fctfi--^? + *'*, X^kR-^' N ^kR^re-dRd^- 

Since ^ must satisfy (4), page 400, the radial component reduces to the 
simpler form 1 

< *-**. 



1 The first thorough investigations of the electromagnetic problem of the sphere 
were made by Mie, Ann. Physik, 26, 377, 1908 and Debye, Ann. Physik, 30, 57, 1909. 
Both writers made use of a pair of potential functions leading directly to our vectors 
M and N. The connection between these solutions and a radial Hertzian vector has 
been pointed out by Sommerfeld in Riemann- Weber, " Diff erentialgleichungen der 
Physik," p. 497, 1927. 



416 SPHERICAL WAVES [CHAP. VII 

To obtain explicit expressions for the vector wave functions M and N, 
we need only carry out the differentiation of (1) as required by (8) and 
(9). The time factor is split off by writing M = me-*" 1 , N = ne-* 1 , 
and we find 

(11) m = + L Zn(kR ) P?(co S 0) n m* i, - z n (kR) 



+ kK-e m [R z 

7.12. Integral Representations. The wave functions 1, m, and n can 
be represented as integrals of vector plane waves such as (18) to (20), 
page 395. For the functions of the first kind, which are finite at the 
origin, we have by (60), page 410, 

_ n p2r f*Tr 

(13) /(i) = \ \ e ikR COB * P(cos a) c . os mp sin a da dp. 

, 4?r Jo Jo sin 

First, we calculate the components of the gradient; namely, df/dR, 

_i -L, ~ -- i, differentiating under the sign of integration. Next we 
R dd R sin d<t> 

observe from an inspection of Fig. 70, page 407, that if the vector k(, 0) 
is directed along the line OQ and R(0, <#>) along the line OP, then 

k . ij = k cos 7 == fc[sin sin 6 cos (<t> 0) + cos a cos 0], 

(14) k i 2 = fc[sin QJ cos cos (<t> ft) cos a sin 0], 
k i 3 = fc sin a sin (0 0), 

the unit spherical vectors ii, i 2 , is referring, of course, to the point of 
observation at P. Then it follows without further calculation that 

(15) I(D = ~ I I k(, p)e ikR C08 y P^(cos a) c . os m& sin a da d/9. 

o m * 47T JO Jo S1U 

To find the corresponding representations of m (l) and n (1) we note that 

R X k *= JfeJB{sin a sin (</> /3)i 2 + [sin a cos cos (</> - 0) 

(16) cos a sin 0]i 3 }, 
(k X R) X k = fc 2 #{sin 2 7 ii cos 7[sin a cos cos (< - p) 

cos a sin 0]i 2 + cos 7 sin a sin (< j3)is} 

*nd, upon differentiating according to (8) and (9), obtain 

(17) mi 1 ' = ~ \ I k X R e ikR cos ^ P?(cog a) c . os m/3 sin a da dB, 
^ ' J 4?r Jo Jo sin 



SBC. 7.13] ORTHOGONALITY 417 



e ikR cos y pm( cos a ) COS 

sm 



To obtain the corresponding integral representations of functions of the 
third and fourth kinds the integration over a from to w must be replaced 
by a contour in the complex domain as described on page 410. 

Rectangular components of the vectors 1, m, and n can be calculated 
without great difficulty with the help of these integral representations, 
although the resultant expressions are somewhat long and cumbersome. 
The vector integrands are resolved into their rectangular components 
and become explicit expressions of a, 0, 6, <t>. Thus, for example, 
k x = k sin a cos 0. The recurrence relations for the spherical harmonics 
are then applied to eliminate these factors and to reduce the integrals to 
a form that can- be evaluated by (13). 

7.13. Orthogonality. The scalar product of any even vector with any 
odd vector, or with any vector differing in index m, is obviously orthogonal 
and we need consider only such products as do not vanish when integrated 
over </> from to 2w. Thus 



JT 



where 5 = if m > 0, d = 1 if m = 0. To reduce (19) we apply the 
formula 



___ 0, when n 

2 (n + w)! / 



u 

> when n = 



Integration of (19) over gives, therefore, zero when n 7* ri, or, when 

n = n', 

(21) J J l %w 1^ sin dO d<i> 



A final reduction follows from the recurrence relations (33) and (34), 
page 406, and we obtain the normalizing factor 



418 SPHERICAL WAVES [CHAP. VII 



(22) I Ml, 1, sin dO d<t> 

JO JO O m * o m * 

- n _i_ *\ 2?r (n + m) ! 



(n 



The same formulas lead directly to the integrals 

/2 f* 

(23) I I m, m e sin 
Jo Jo o m 



o m * 



27T 



(24) I I n, -n, sin 

JO JO o mn o m " 

,, , 27r (n + m)! 

= ( 1 -f- o) 7^ : r-rrt 7 \~i ' 

(Zn + 1J (n in) 



1} { (n 



while all products differing in the index n are zero. 
Upon examining the cross products, we find first 

/rtp\ l*i j , i/ii^N 71 " 771C 

(25) I 1, m , d0 = (1 + 5) 2? n ^ ^ ^ - 

JO O mn e tt -^ 8m " " 

hence, 

r* r 2r 

(26) I I l e m , sin d0 d<t> = 
Jo Jo O wn wn 

for arbitrary values of n and n'. Likewise 



(27) 



r c 2 * 

I m, - n , sin dO d<f> = 0. 

Jo Jo O mn e wn 



The complete orthogonality of the system is spoiled, as in the cylindrical 
case, by the product l e n e , whose integral over the sphere does not 

o o 

vanish if n = n 7 . In that case we find 
(28) I I l e -n, sined6d<t> 

JO JO o m * o m * 

f + . ^ 2ir (n + m)! / , 



To complete the orthogonalization one might treat k r as a variable 
parameter and integrate over k f and R as in Sec. 7.2; such a procedure 
usually proves unnecessary. 

7.14. Expansion of a Vector Plane Wave. To study the diffraction 
of a plane wave of specified polarization by a spherical object one must 



SEC. 7.14] EXPANSION OF A VECTOR PLANE WAVE 419 

first find an expansion of the incident vector wave in terms of the spherical 
wave functions 1 , m 6 , n e . 

o mn ^mn Q mn 

Consider the vector function 

(29) f (z) = a e ik * = a e ikR c <* ', 

where a is an amplitude vector oriented arbitrarily with respect to a 
rectangular reference system. Let a be resolved into three unit vectors 
directed along the x~, y-, z-axis respectively. Then 

a* = sin 6 cos < ii + cos cos 4* i% sin < is, 

(30) Oy = sin 6 sin < ii + cos sin < i 2 + cos <t> i 3 , 
a* = cos 6 ii sin 6 i 2 , 

where ii, i 2 , ia are again the unit vectors defined in Fig. (8), page 52, 
for a spherical coordinate system. Now the divergence of the vector 
functions & x exp(ikz) and a^exp^fcz) is zero and consequently they may 
be expanded in terms of the characteristic functions m and n alone. At 
R = the field is finite and we shall, therefore, require functions of the 
first kind. It is apparent, moreover, that the dependence of (30) on 
limits us to m = 1; whence upon consideration of the odd and even 
properties of (11) and (12), we set up the expansion 

(31) a 

To determine the coefficients we apply the orthogonality relations of the 
preceding section. 

(32) f f a, m^ 1 ^ e ikR coa e sin 6 dO d<t> = 2iri n n(n + 
Jo Jo 

whence by (23), 



Likewise 

(34) I I a x n<p n e* R coa e sin dd d<t> 
Jo Jo 



which in virtue of (24) gives 

b = > 
On l 



n(n 
hence, 



(36) W = 

n = 



420 SPHERICAL WAVES [CHAP. VII 

The same process gives for the wave polarized in the ^/-direction the 
expansion 

(37) a* 



Since the longitudinal wave function a z exp(ifc2) has a nonvanishing 
divergence, its expansion must involve the 1 functions. It turns out, in 
fact, that only this set is required and one finds without difficulty that 



(38) a,e<* = 

n=0 

Problems 
1. Show that the spherical Bessel functions satisfy 



0, when m j n, 



2 2n + 1 



provided n and m are integers satisfying the conditions n ^ 0, m > 0. For the appli- 
cation of such integrals to the expansion of functions in Bessel series see Watson, 
"Bessel Functions," page 533. 

2. Show that the cylindrical Bessel functions satisfy 



and with the help of these formulas show that the spherical Bessel functions satisfy 



d* /cos x\ I 1 d n / cos 



3. Show that if R 2 = R% + R\ 2R Q Ri cos y, 

sin M sin kRo sin < 



l-i kR 

sin fc# cos fe/Zi K> < 7? 

cos A;J2 fcfto A;J?i 
o(cos y) =* 

1 jf ^ jjj^ 

4. Show that in the prolate spheroidal coordinates defined on page 56 the equation 



PROBLEMS 



421 



assumes the fonr 



0. 



Let 



), and show that the above separates into 

- C 



- 1 



+ 



1 - 
'0, 



-C 



0, 



where m and C are separation constants. Next show that a set of prolate spheroidal 
wave functions can be constructed from 



sin 



in which both the "angular functions" Se and the "radial functions" Re satisfy the 
equation 

(1 - zV - 2(m + l)zt/' + (6 - cs)y = 0, 



with m an integer or zero, b C m, and c = fed = 2ird/\. 

6. When the wave functions of Problem 4 apply to a complete spheroid one 
must choose m = 0, 1, 2, . . . , and find the separation constant b such that the 
field is finite at the poles r/ = 1. A discrete set of values can be found in the form 

6 n - n(n + 1) +/ n (c), n = 0, 1, 2, .... 

Show that the functions Se and Re can be represented by the expansions 



Se l mn (c, z) = d.TT(i), T?(z) = (1 - * >.% 

S 

where PT +m is an associated Legendre function and the prime over the summation sign 
indicates that the sum is to be extended over even values of s if n is even and over 
odd values if n is odd. 



e l mn (c, z) 



._ (2m +*)! 

*' n ; d a j a . 

s\ 



where jt+m(cst) is a spherical Bessel function. The independent solution is obtained 
by replacing j,+ m (cz) by n s + m (cz). Show that the coefficients satisfy the recursion 
formula 



8+2 



- 1) 



(s +2m +2) (a -f 2m + 1) 

(2s + 2m + 5)(2 -f 2m + 3) " 1 "" 1 " ' (2s + 2m - 1)(2 -f 2m - 3) 
, f 2s 2 -f 2(2m + 1) + 2m - 1 



*~ f 



+ (a -f 2m + 1) 



- b \d. 



L(2s + 2m - 1)(2 + 2m + 3) 
and from this relation give a method of determining the characteristic values 



422 SPHERICAL WAVES [CHAP. VII 

6. Prove that the spheroidal functions defined in Problem 5 satisfy the relations 



Se l mn (c, z) - X e**(l - t*) m Se l mn (c, f) dt 



where X n is one of a discrete set of characteristic values. 

7. Write the equation VV 4" &V in oblate spheroidal coordinates and express 
the solutions in terms of the functions Se and Re denned in Problem 5. 

8. A system of coordinates , 77, < with rotational symmetry is denned by 

ds* = hi d? -{- h 2 dip + r 2 d<t>* 

where r is the perpendicular distance from the axis of rotation. If the field has the 
same symmetry as the coordinate system, its components are independent of <. 
Show that the equations 

V X E - ico/iH =0, V X H + (to* - <r)E * 
break up into two independent groups: 

iunHt = 0, 



(I) 4" Tt (rE<t>) 



~ (htH,) - . (hlH i) 4. (iw - 
ihtldt di\ J 



T 5 (/l2 ^ } ~ I" (/ll ^ } - 
ii/iaL 6 ^ ^ J 



Show that (II) is satisfied by a potential Q such that 

n - a 

Q - 






5^ \r/ii 5^/677 \rfc, ^r? / r V 

(I) can be integrated in the same manner. (Abraham) 

9. Apply the method of Problem 8 to find a rotationally symmetric electro- 
magnetic field in prolate spheroidal coordinates. In the case of electric oscillations 
directed along meridian lines show that Q satisfies 



vW - 0, 

which separates into 

(? - DQ" + (dW? - OQi - 0, 

a - ^e;' + (~dw + C)Q, - o, 



PROBLEMS 423 

where Q Qii&QM. Note that both of these are a special case of the equation 

(1 z z )y" 2(a + l)zy' + (b c 2 z 2 )y = 
when a 1. Show that the field is given by 

ico/i Qi dQz 



Icon 



, , 

Pfc V (1 - l')(' - I 2 ) dt 

10. Two types of spherical electromagnetic waves have been discussed in this 
chapter: the transverse electric for which ER = 0, and the transverse magnetic for 
which HR 0. Obtain expressions for the radial wave impedances as was done 
previously for cylindrical waves. 

11. Prove the expansions (86) and (87) of Sec. 7.10 and show that when 0i = ir, 
Ri QO, Eq. (87) goes over asymptotically into the expansion of a plane wave. 

12. Obtain expressions for the rectangular components of the vector spherical 
wave functions 1, m, and n by the method suggested at the end of Sec. 7.12. 



CHAPTER VIII 
RADIATION 

In the course of the past three chapters we have studied the propaga- 
tion of electromagnetic fields with no concern for the manner in which 
they are established. We consider now the sources, and the fundamental 
problem of determining the intensity and structure of a field generated 
by a given distribution of charge and current. 

THE INHOMOGENEOUS SCALAR WAVE EQUATION 
8.1. Kirchhoff Method of Integration. Mathematically the problem 
of relating a field to its source is that of integrating an inhomogeneous 
differential equation. Let \l/ represent a scalar potential or any rec- 
tangular component of a field vector, and let g(x, y, z, t) be the density of 
the source function. We shall assume that throughout a domain V the 
medium containing the source is homogeneous and isotropic, and that 
its conductivity is zero. The presence of conductivity introduces serious 
analytical difficulties which can be avoided in most practical applications 
of the theory. The effect of conducting bodies in the neighborhood of 
oscillating sources will be treated as a boundary-value problem in Chap. 
IX. Subject to these restrictions, the scalar function \l/ satisfies the 
equation 

(i) W-JI^T = -0foy,M), 

where v = (CM)""* is the phase velocity. 

The Kirchhoff theory of integration is an extension to the wave 
equation of a method applied in Sec. 3.3 to Poisson's equation. Let V 
be a closed domain bounded by a regular surface S and let < and \l/ 
be any two scalar functions which with their first and second derivatives 
are continuous throughout V and on S. Then by (7), page 165, 



f* - * fj) 



(2) <*v V - *Y*) * = s * - * da, 

where d/dn denotes differentiation in the direction of the positive, or 
outward, normal. Let (x', y' , z r ) be a fixed point of observation within 
y and 



(3) r = V(x' - x)* + (y f - y) + (*' - z) 2 

the distance from a variable point (x, y, z) within V or on S to the fixed 

424 



SBC. 8.1] KIRCHHOFF METHOD OF INTEGRATION 425 

point. For < we shall choose the spherically symmetric solution 

(4) * = J 

of the homogeneous equation 



where / It + - 1 is a completely arbitrary analytic function of the argu- 
ment t -\ 

As in the analogous problem of the stationary field, < has a singularity 
at r 0; this point must, therefore, be excluded from the domain V 
by a small sphere Si of radius r\ drawn about (re', y', z') as a center. The 
volume V is now bounded externally by S and internally by Si. Further- 
more, if we denote the left side of (2) by / and make use of (1) and (5), 
we obtain 

(6) 

Or, since 



Upon integrating over the entire duration of time this gives 

< 8 > r. j * - - r. * j> * + ? i (* ss - * 2) * 

The next task is to choose the function f(t + - J in such a way that 

the last term of (8) vanishes. To this end let V = r/v and take for 
f(t + t') the impulse function 

~+O 

(9) f(t + t')=--= e ^ = S + O, 

defined by (35), page 291, with the property 
(10) 



To avoid any question as to the continuity of <t> and its derivatives, we 
shall imagine d to be exceedingly small but shall not pass at once to the 
limit 5 = 0. Thus defined, St(t + V) vanishes everywhere outside an 
infinitesimal interval in the neighborhood of t = V. With this in mind 



426 RADIATION [CHAJ>. VIII 

it is apparent that if F(f) is an arbitrary function of time, then 

(11) F(-O = M S(t + t')F(t) dt, 



and that df/dt approaches zero with vanishing 5 for all values of t. 

Since now <t> and d<t>/dt vanish at the limits t = <*> and t = oo 1 for 
all finite values of r, it follows that the last term of (8) must likewise 
vanish and, consequently, that 



r idt= - ( dv r s<>(t + 1'} g(x > y > *> 

J- * JV J ~ r 



When equated to the right-hand side of (2), this leads to 



Over the surface Si we know that d/dn = d/dr, so that 
(14) 



I / ^ 
J ^^~ 




Passing to the limit as r\ > 0, this reduces to 



and hence by (11), 

C / z.t. ;JA\ 

Ida = - 



1 Obviously these limits may now be drawn in to enclose an infinitesimal interval 
containing the instant t = r/v. 



SEC. 8.11 KIRCHHOFF METHOD OF INTEGRATION 427 

Substituting this value into (13) gives 



(17) *(*', y>, tf, 0) = - ^ ff(a . f p> 2, _ o dt , 

+JL r r [i^+o^-^ 

47rj,sJ_aoLr dtt dn r 

To reduce these integrals to a simpler form we note that 

(18) ^[Mfl]. w + 0f ^) + i*. &( , 

and that 

d du 1 3r a5o 



An integration by parts gives 

tdS (t + t') lldr 



and application of (11) leads to 



(21) *(*', j/', ', 0) = 



1 f [i (w\ d (i\ 

S J s b We- -r " ^ W 



According to (21) the value of ^ at the point x' y y' ', 2;' and the instant 
i = is obtained by summing the contributions of elements whose phases 
have been retarded by an amount t' = r/v. But the location of the 
reference point on the time axis is purely arbitrary; consequently, the 
Kirchhoff formula becomes 

(22) *(x',y',zr,() = J 



in which the symbol 

(23) fo 

will denote a function with retarded phase. 

The volume integral in (22) is a particular solution of the inhomo- 
geneous wave equation representing physically the contribution to 
^(z', t/ 7 , z', t) of all sources contained within V. To this particular solu- 
tion is added a general solution of the homogeneous equation (5) expressed 



428 RADIATION [Cii^p. VIII 



as a surface integral extended over S and accounting for all sources 
located outside S. In case the values of \l/ and its derivatives are known 
over S, the field is completely determined at all interior points, but it is 
no more permissible to assign these values arbitrarily than in the static 
case. We shall have occasion below to discuss this point further in 
connection with the Huygens principle. 

The behavior of ^ at infinity is no longer obvious since r enters 

explicitly into [g] through the retarded time variable t --- The field, 

however, is propagated with a finite velocity; consequently, if all sources 
are located within a finite distance of some fixed point of reference and if 
they have been established within some finite period in the past, one may 
allow the surface S to recede beyond the first wave front. It lies then 
entirely within a region unreached by the disturbance at time t and 
within which \[/ and its derivatives are zero. In this case 



= 1 \g (x, y, z, t - 



(24) *(*', y', z', t) = g x, y, z, t - dv, 

where V now represents the entire volume occupied by sources. 

The reader has doubtless noted that the choice of the function 

/ It + - ) is highly arbitrary, since the homogeneous equation (5) admits 

also a solution - / f t -- V This function leads obviously to an advanced 

time, implying that the quantity \l/ can be observed before it has been 
generated by the source. The familiar chain of cause and effect is thus 
reversed and this alternative solution might be discarded as logically 
inconceivable. However the application of "logical" causality prin- 
ciples offers very insecure footing in matters such as these and we shall 
do better to restrict the theory to retarded action solely on the grounds 
that this solution alone conforms to the present physical data. 

8.2. Retarded Potentials. When (24) is applied to the scalar poten- 
tial and to the rectangular components of the vector potential we obtain 
the formulas 

(25) 

(26) 

which enable one to calculate the field from a given distribution of charge 
and current. Here x f represents the coordinate triplet #', y f , z r } and x 

T 

the triplet x, j/, z, while the retarded time is expressed by t* = t --- 



SEC. 8.2] RETARDED POTENTIALS 429 

This statement, however, is subject to one condition. We have in 
fact only shown that (25) and (26) are solutions of the inhomogeneous 
wave equation. In order that they shall also represent potentials of an 
electromagnetic field it is necessary that they satisfy the relation 

(27) V A + M ^ = 0. 

ot 

The prime attached to the operator V denotes differentiation with 
respect to x 1 ', y r , z' at the point of observation. 

We shall show that this condition is fulfilled by (25) and (26) provided 
the densities of charge and current satisfy an equation of continuity. 
Since d/dt = d/dt* at a fixed point in space we have 

- 



Now when the operator V is applied to any power of r, one may write 

T 

Vr n = V'r n . The variables x' occur in J(x, t*) only through t* t ; 

consequently, 

On the other hand 

1 r)T 

(31) V J = - -^ Vr + (V J)i*_ constant, 

so that 

(32) V J = (V J),*_ constant - V J. 

Equation (29) can now be written 

(33) V' A = - J V r Jj dv + ; I J (V - J>-constant * 

By the divergence theorem 

(34) f V ( i J dv = I J n da = 0, 

JF \r/ Jsr 

since the closed surface S bounding V can be taken so large that J is 
zero everywhere over S. Then on combining (28) and (33), we find 

(35) i T '.A + .-^ 



430 RADIATION [CHAP. VIII 

and this must vanish since the conservation of charge requires that 

(36) (V J),._ constant + ~j* = 

at any point whose local time is t*. 

8.3. Retarded Hertz Vector. By virtue of (36) it is sufficient to 
prescribe the current distribution in space and time. The charge density 
can then be calculated and the potentials determined by evaluating the 
integrals (25) and (26). However, in most cases it is advantageous to 
derive the field from a single source function by means of a Hertz vector. 

Let us express the current and charge densities in terms of a single 
vector P defined by 

(37) J = ^ P = -V-P. 

The equation of continuity is then satisfied identically and the field 
equations are 

(I) V X E + IL ^ = 0, (III) V H = 0, 

PXH- ^ =^, (IV) V-E= -iv-P. 



6 



As shown in Sec. 1.11, these equations are satisfied by 

(38) E = VV n - /ze ^5, H = eV X -~; 

at" Ov 

where n is any solution of 

(39) v x v x n w n + /x^ -^ ~~ 7 

The rectangular components of II , therefore, satisfy the inhomogeneous 
wave equation 

(40) V 2 n y - ju ^Jr = -\ P *> U = *> 2j 3) * 

It need scarcely be said that the vector P defined by (37) is not 
identical with the dielectric polarization defined first in Sec. 1.6. The 
(x ,x ,# 3 ) vector P of the present section measures 
^ the polarization, or moment per unit vol- 

ume, of the free-charge distribution and 
consequently is not equal to D e E. 

To facilitate the calculations that fol- 
low a small change in notation will be 

^ , . made. The point of observation in Fig. 

Fio. 71. A current element is TIT, / .c j 

located at 1, 2 , g, while x it xt, x 8 71 whose radial distance from a tixed ori- 

locate a fixed point of observation. gm Q is R will now be located by the rec- 

tangular coordinates (x\ 9 z 2 , 3). The current element located a distance 




SBC. 8.4] DEFINITION OF THE MOMENTS 431 

Ri from has the coordinates (1, 2 , 3) and its distance from the point 
of observation is r, so that 




(41) r = 

We need consider only the harmonic components of the variation, from 
which general transient or steady-state solutions may be constructed. 
Let 

(42) J = J () e-*, p = po() 6-*", po = ^ V J (a 

(43) P-PoCQe-*', Po = ^Jo. 

CO 

Applying (24) to (40) and noting that in the present case k = co \/M* = co/v, 
we obtain the fundamental formula 

(44) n(x,0 



A MULTIPOLE EXPANSION 

8.4. Definition of the Moments. To evaluate the retarded Hertz 
integral one must frequently resort to some form of series expansion. 
The nature of this expansion will be governed by the frequency and the 
geometry of the current distribution. In the present section we shall 
consider an extension to variable fields of the theory of multipoles set 
forth in Sees. 3.8 to 3.12 and Sees. 4.4 to 4.7 for electrostatic and mag- 
netostatic fields. 

According to (87), page 414, when R > RI, the integrand of (44) above 
can be expanded in the scries 

oo 

(]\ _ 7 'k ^v (2-n 4- "HP /Vn ^ i" (kJ?-,} Jj^VfrTPV 

\-*- / vtv s^t \ill |^ **)*- nvkUo // J7t\'^-*i'l/ '^n \li>JLvJ , 

consequently, if R is greater than the radius of a sphere containing the 
entire source distribution, 

f9h T\(r t\ f>i<*t ^^ fO-vj _1_ 1 ^7)(l)/'^/?^ I l$ n (t\i (Irft^P (nr\<z<v} dii 

\AJ J.JL^l/, 1>J ~ t> ^ \^tll ^^ .Jfl n \^t\/J.ifJ I X^QVS / Jn\iv-LvIJ J. ri\^'-'o I ) Wi/. 



Suppose now that PU(?) is expressed as a function of 72i, 7, 0, where </> 
is the equatorial angle about an axis drawn from through the point of 
observation. Then the vector PoCRi, 7, <) can be resolved into scalar 
components and each component expanded in a series of spherical 



432 RADIATION [CHAP, VIII 

harmonics as in (36), page 406. In virtue of the ortnogonality of the 
functions P n (cos 7), the integral on the right reduces essentially to an 
expansion coefficient times [j n (kRi)]*. The success of this attack will be 
determined by the nature of the current distribution and the difficulties 
encountered in computing the expansion coefficients. 

If the wave length of the oscillating field is many times greater than 
the largest dimension of the region occupied by current, the series (2) 
can be interpreted in terms of electric and magnetic multipoles located at 
the origin. We shall assume that for all values of R\ within V 



(3) Jfcfli = 1, 

where X is the wave iength in the medium defined by and /i. Conse- 
quently the function j(kRi) can be replaced by the first term of its 
series expansion. By (25), page 405, 

(4) 

The total field will be represented as the sum of partial fields by 

e 

(5) n(x,f) = 2n"CM), 

n-0 

and, hence?, 



* *71 

(6) n<">(*, = 4- ^ **+'*-<" A>(0 J^ Pott) B! Pn(cos 7) dv. 

Since P (cos y) = 1 and htf*(kR) = ~"jD e ** we ^ ave at once ^ or 
the first term 

1 fikR-i<*t C 

(7) n"> = ^ P "> 5-g-, P< = J r Po({) dv. 

Let $ be any scalar function of position. By the divergence theorem 

(8) f V . tyPo) dv = I r^Po n da, 
Jr J5 

and, hence, 

(9) I Po V^ dt; * - I ^V Po + I $Po n da. 

If S encloses the entire source distribution, P n will be zero over $ 
and the surface integral will vanish. Choose now for ^ a rectangular 
component , of the radius vector R t . Then 



(10) 



- f *, 



SBC. 841 DEFINITION OF THE MOMENTS 433 

consequently 

(11) P (1) = f Po<to= f R lpo <fo. 

Jv Jv 

The partial field n (0) is generated by an oscillating electric dipole whose 
moment p (1) is defined exactly as in the electrostatic case, Eq. (47), page 179. 
To obtain the second term of (6) we note that Pi(cos 7) = cos 7, 

" Hence ' 



/1O\ ww 1 1 \ *''*' I * i ' 1 -1.0 j I ^_ 

(12) n< l > = ~ (- + -_) e ^--' P ({) R, cos 7 dw. 

4:7T \/I A:/t/ / J^ 

To interpret the integral we first expand the integrand. 

(13) P Ki cos 7 = (Ri R) ^r 

1 



27e l x p o) x R + p o(Ri R) + Ri(R Po)]. 

The magnetic dipole moment was denned in (28), page 235, as 






m< = R! X J 



o 



Since R is independent of the variables { of integration, one has, therefore, 

(15) ^ I (Ri X Po) X R dv = -^m' 1 ' X R. 
*K J v co/t 

The electric quadrupok moments of a charge distribution were defined 
in (48), page 179, as the components of a tensor, 

(16) Pi, = I t ,p cfo = - f J t {,V - Po dv. 

Jv Jv 

Upon putting ^ = { t {/ in (9) this gives 

(17) p./ = J ({,P * + Po,) di; = p,i. 
Furthermore, 

(18) Ri-R- 



Consequently, 

(19) ~ ( [P(Ri R) + {.(P. . R)] dv 



1 dR 1 ,_ 

2 = 



434 



RADIATION 



[CHAP. VIH 



The quantities pW are the components of a vector 
(20) p (2) = 2 p VB, 

where 2 p is the tensor whose components are p^. [Cf. Eqs. (23), (24), 
page 100.] 

The partial field n (1) represents the contributions of a magnetic dipole 
and an electric quadrupole. 




Comparison of (21) with (7) shows that the two fields differ in the first 
place by a factor A/M*, or approximately 3 X 10 8 , and consequently the 
moments m (1) and p (2> must be very much larger than p (1) if II (1) is to 
be of appreciable magnitude. On the other hand, it must be remembered 

that the electric moments of a cur- 
rent distribution are functions of the 
frequency as well as of the geometry 
[Eq. (43), page 431]. If p (2) is ex- 
pressed in terms of current rather 
than charge, we see that the magni- 
tude of n (l) is independent of fre- 
quency, whereas the magnitude of 
n (0) diminishes as 1/co with increas- 
ing frequency. 

The remaining terms of the series 
in n (n) represent the contributions of 
electric and magnetic multipoles of 
higher order. These must be ex- 
pressed in terms of tensors and the 
labor of computation increases rap- 
idly. The magnitudes of the successive terms progress essentially 
as (o>/c) n . In case the rate of convergence is so slow that higher order 
terms must be taken into account, one is usually forced to adopt some 
other method of calculating the radiation. 

8.6. Electric Dipole. Let us suppose that the charge distribution is 
such that only the electric dipole moment is of importance. This is the 
case, for example, of a current oscillating in a straight section of wire 
whose length is very small compared to that of the wave. The coordinate 
system is oriented so that the positive z-axis coincides with the dipole 
moment p (1) . The vector p (1) can be resolved into its components in a 
spherical system as illustrated in Fig. 72. 




FIG. 72. Components of field about an 
electric dipole. 



(22) 



O> = 



cos i 



p w sin 9 i s , 



SEC. 8.5] ELECTRIC DIPOLE 435 

where ii, i 2 , is are the unit spherical vectors designated in Fig. 8, page 52. 
According to (7) the Hertz vector n<> is parallel to p< 1} and can be 
resolved in the same manner. The field vectors are now calculated from 
the formulas (38), page 430, which in the case of a harmonic time factor 
become 

(23) E = vv n + fc 2 n, H = -tcoev x n. 



The differential operators, expressed in spherical coordinates by (95), 
page 52, are applied to the vector 

(24) n<> = n<> cos ii - n sin Oi 2 . 

If Ro S3 vR is a unit vector directed from the dipole towards the observer 

r> 

and t* = t -- i then 
v 



(25) E<> = *-*" -p [3R (R p (1) ) - p<] 

fif ^ 2 1 

- [3Ro(R p>) - p<] - J [R X (Ro X p<)]|, 

(26) H --^ 



The general structure of a dipole field is easily determined from these 
expressions. We note in the first place that the electric vector lies in a 
meridian plane through the axis of the dipole and the magnetic vector is 
perpendicular to this plane. The magnetic lines of force are coaxial 
circles about the dipole. At zero frequency all terms vanish with the 
exception of the first in l/R 3 in (25), which upon reference to (27), 
page 175, is seen to be identical with the field of an electrostatic dipole. 

Moreover, if we replace p (1) by the linear current element - / ds according 

to (43), page 431, it is apparent that the first term of (26) can be inter- 
preted as an extension to variable fields of the Biot-Savart law, (14), 
page 232. The terms in (25) and (26) have been arranged in inverse 
powers of R and the ratio of the magnitudes of successive terms is kR, 

or 2*- In the immediate neighborhood of the source the "static" and 
A 

"induction" fields in 1/K 3 and 1/R 2 predominate, while at distances such 
that R X, or kR 1, only the " radiation " field 

(27) 



need be taken into account. At great distances from the source, measured 
in wave lengths, the field becomes transverse to the direction of propagation. 



436 RADIATION [CHAP. VIII 

If we let R oo and at the same time increase p (1) so that the intensity 
of the field remains constant, we obtain in the limit the plane waves 
discussed in Chap. V. 

This transverse property of the wave motion at great distances is 
common to the fields of all electromagnetic sources, but we must not 
overlook the fact that in the vicinity of the source there is a longitudinal 
component in the direction of propagation. The radiation field of an 
electric dipole, for example, vanishes along the axis of the dipole, but for 
sufficiently small values of R there is an appreciable longitudinal com- 
ponent of E (0) of the form 



This component, and not the radiation field, determines the distribution 
of current in a vertical antenna above the earth. 

Let us calculate next the energy flow within the field. For this 
purpose we shall make use of the complex Poynting vector derived in 
Sec. 2.20. If S is the mean value of the energy crossing unit area in the 
direction of the positive normal per second, then 

(29) S = Jfe(S), S* = iE X H. 

To calculate the complex Poynting vector S*, we first write (25) and 
(26) in component form. 



From these we construct 
(31) S* = 

and obtain 



i - 
R* l 32rr 2 e 



Next the normal component of S* is integrated over the surface of a 
sphere of radius R. 



(33) S2 fl' sin 9 d6 d<t> 



SBC. 8.6] MAGNETIC DIPOLE 437 

Now by (29), page 137, the real part of (33) gives the mean outward flow 
of energy through a spherical surface enclosing the oscillating dipole. 
Since this real part is independent of the radius of the sphere, it must be 
the same through all concentric spheres and we discover that the system is 
losing energy at a constant rate 



(34) TF<> = - /x \/^ |p (1) | 2 watts. 



The quantity W measures the radiation loss from the dipole. In the 
expansion of an electromagnetic field, only those terms that vanish at 
infinity as R~ [ give rise to radiation. In the present instance there is, 
of course, an instantaneous energy flow associated with the other terms 
of (25) and (26), but the time average of this flow over a complete cycle is 
zero. The terms in R~ 2 and R~* account for the energy stored in the 
field, energy which periodically flows out from the source and returns to 
it, without ever being lost from the system. 

In free space AC O = 4?r X 10~ 7 , \/Moeo = i X 10~ 8 , and hence 
(35) W <> = J X 10- 15 co 4 |p< 1} l 2 = 173 X 10- 16 v VT watts 

where v is the frequency. The radiation increases very rapidly with 
increasing frequency and decreasing wave length. 

8.6. Magnetic Dipole. The current distribution may be such that 
only the magnetic dipole moment m (1) is of importance. At sufficiently 
low frequencies this is certainly the case for any closed loop of wire. Then 



The components of the field calculated from (23) are 



(37) W$ = Q-, - Jj) cos 6 \mW\e-"*, 

ik 



The structure of this field is identical with that of the electric dipole but 
the roles of E and H are reversed. The vector H (1) lies in a meridian 
plane through the dipole and has a radial as well as a transverse com- 
ponent. The lines of E (1) are concentric circles about the dipole axis. 
When k = 0, (37) reduces to the expressions derived in Sec. 4.5 for the 
static field of a fixed magnetic dipole. 

To calculate the radiation we construct the complex Poynting vector 
as in (31) and integrate its radial component over a sphere of radius R. 



438 RADIATION [CHAP. VIII 

Only the terms of (37) in 1/R contribute to the result and we obtain 
(38) WM = ^ ^ |m (1) | 2 watts. 

In free space VVoAo = 376.6 ohms, in which case 



fjW) TF (1) = 10& 4 lw (l) l 2 watts. 

W^/ l I 

For the sake of example consider an alternating current in a single 
circular loop of radius Ri meters. The maximum value of the current is 
7 and the wave length is assumed to be very much greater than fii. 
The phase of the current is then essentially the same at all points of the 
circuit, whence it follows from (43), page 431, and (11), page 433, that 
the electric dipole moment is negligible. The magnetic moment 
|w (1) | = TTJBf/oJ consequently, the energy radiated per second is 

(40) 

The radiation resistance (R is defined as the equivalent ohmic resistance 
dissipating energy at an equal rate. Since Jo represents the maximum 
current amplitude, 

(41) W = ifltfj 
and, hence, in the present instance 

/R \ 4 (R Y 

(42) (R = 3207T* ( -^ I = 3.075 X 10 5 ( ~ ) ohms. 

\ A/ \ A/ 

This result is restricted in the first place by the condition that the second 
term in the expansion of ji(kRi) shall be negligible with respect to the 
first. 

(43) 

and hence ji(kRi) can be approximated by the first term if kRi < 1. 
It is in fact a general rule that these functions can be represented by their 
first term as long as their argument is not greater than their order. If 
then X = 10 JRi, we find the radiation resistance to be about 30 ohms. 
As X decreases, (R increases very rapidly, but the electric dipole and 
quadrupole moments now begin to play a part. On the other hand if 
the wave length increases, it is clear that radiation soon becomes entirely 
negligible. Currents whose wave length is long compared to the largest 
dimension of the metallic circuit give rise to no appreciable radiation and 
are said to be quasi-stationary. 

RADIATION THEORY OF LINEAR ANTENNA SYSTEMS 

8.7. Radiation Field of a Single Linear Oscillator. The rigorous 
determination of the current distribution in a conductor is a boundary- 



SEC. 8.7] 



LINEAR OSCILLATOR 



439 



value problem. A steady-state solution of the field equations must be 
found, corresponding to a given impressed e.m.f., which satisfies the 
necessary conditions of finiteness at interior points of the conductor and 
continuity of the tangential components of field intensity across its 
surface. This problem has never been completely solved for a finite 
length of a cylindrical conductor, but a good idea of what takes place may 
be gained from a study of a conducting, prolate spheroid whose eccen- 
tricity differs but little from unity. Such an investigation was made first 
by Abraham, 1 and a number of papers have dealt more recently with 
the same subject. All this work confirms what one would naturally 
expect : the current distribution along an iso- 
lated straight wire is essentially harmonic and 
the ratio of antenna length to wave length is 
half an integer, as long as the cross-sectional 
dimensions are very small with respect to 
the length. A much greater source of error 
than the finite diameter of the conductor is 
the damping due to radiation. As the an- 
tenna current flows outward from the feeding 
point towards the ends, energy is dissipated 
in heat and radiation. Consequently the 
assumption of a constant amplitude at the 
current loops along the wire is incorrect. 
Superimposed on a standing antenna current wave is a damped traveling 
wave supplying the losses. The effect of this modified distribution on 
the radiation pattern may be considerable. 

The current distribution is also affected by proximity to the ground 
or other objects. The trend in contemporary antenna design is towards 
systems of linear antennas supported some distance above the earth and 
in this case the effect of the earth on current distribution can usually be 
neglected. When, on the other hand, a structural steel tower is used as 
an antenna, the capacity to ground may have an important influence on 
the distribution. 

In the present section we shall consider the radiating properties of 
linear antennas completely isolated in free space. The manner in which 
the radiation is reflected and refracted by the earth will be taken up in 
the following chapter. We shall assume that the current has been 
measured at a number of points along each conductor and that this 
measured distribution can be represented by one or more harmonic 
standing or traveling waves. 

Let us consider first a single straight wire of length I as shown in 
Fig. 73. The current at any point measured from the center of the 

1 ABRAHAM, Ann. Physik, 66, 435, 1898. 




Single linear oscillator. 



440 RADIATION [CHAP. VIII 

wire is 7 = u()exp( iwf). Then by (44), page 431, the Hertz vector 
of the distribution is 



where d is an element of length along the conductor. The radiation 
field of an element d is by (27), page 435, 



(2) 



so that upon integrating the contributions of all elements along the length 
of the wire we obtain 



^ J- < 



(3) Et -- 

Now if ry> I one may consider sin 6/r to be a slowly varying func- 
tion of . This means that the radius vectors from current elements to 
the point of observation are essentially parallel, so that sin 6/r may be 
replaced by sin 6/R and removed from under the sign of integration. 
This approximation is the more nearly exact the larger 72, and since the 
radiation is the same through all concentric spheres about the origin 0, 
we may take R as large as we please. The phase, however, must be 
considered with more care. From the figure it is clear that the time 
required for a wave emitted by the element at to reach a distant point 
is less than that required for a wave originating at by the amount 
cos 6/c, and consequently their phase eventually differs by fc cos 6, 
since fc = 27T/X. It follows that, as R > oo , the field intensities approach 
the limit 

//ZQ jj IOJ/ZQ sin 6 

(4) ^ aa ^^*- 47 ~7T ( 

A sinusoidal current distribution is represented by 

(5) u() = 7 sin (& a), 

and the condition that the current shall be zero at the ends = 1/2 is 
satisfied by 

(6) M() = Jo sin mw U + ^ j, * = -y 

The integer m is equal to the number of half wave lengths contained in 
the length Z, and 7 is the current amplitude at a loop position. The 
integral (4) is readily evaluated and leads to 



Sue. 8.7] LINEAR OSCILLATOR 441 



< 7 > *- -c 

The numerical factor is 



hence, 

(mw \ 
cos I ~2~ cos ) fljjj..^ 

(9) Eo = -i607o ^-r ; e p , when w is odd, 

sin t/ / 

or 



. /* 

sm 1 - 



rraTr 
-^r- COS 



(10) EQ ~ 60/o 7 ^ ^ ^ when m is even. 

sin (/ /e 

The expressions do not depend on the length of the radiator, but only 
upon the mode of excitation. The field is that of a dipole located at the 
center whose amplitude is modified by a phase factor F(6). 

The intensity of radiation in any given direction is again expressed in 
terms of the complex Poynting vector which is radial and whose magni- 
tude is S* = 



j 2 cos 2 [ -^ cos 6 

(ii) 25rft2 



p sin 2 ( cos 6 

s * = o- -ol ^-r- ^' (m = 2, 4, 6, - - - i. 

2?r /2 2 sin 2 ^ v ' ' ' y 

From the location of the zeros and maxima of these functions one can 
gain a fair idea of the radiation pattern. When m is odd, the zeros of 
S* occur at 

/-*^\ * 1 3 771 

(12) cos 6 = ,,... , 

m m m 

while the maxima are found from the relation 

(13) ~- tan 6 sin 6 = cotan ( -^ cos 0V 

Likewise, when m is even, the zeros occur at 
/n> * ^ 2 4 

(14; COS = 0, ;;; j 

mm m 

and the maxima correspond to angles satisfying 

(15) -Tj- tan sin = tan ( -^- cos 6 V 

The radiation patterns of the first four modes are shown in Fig. 74. 
In each case the current distribution is indicated by a dotted curve 



442 



RADIATION 



[CHAP. VIII 



along the radiator, which is broken up into oscillating current sections 
alternately in phase and 180 deg. out of phase. The interference action 
of these current sections gives rise to lobes in the radiation pattern equal 
in number to that of sections m. Between lobes is a radiation node. 




FIG. 74a. m = 1. 





FIG. 74c. m = 3. FIG, 74d. m = 4. 

Fia.74. Radiation patterns of a linear oscillator excited in the modes m = 1, 2, 3, 4. 

The radiation intensity is greatest along the axes of the lobes lying 
nearest the radiator and as m > oo the radiation is directed entirely 
along the wire. Energy can be propagated along an infinite conductor 
but there is no outward radial flow. We shall see later that a Joule heat 
loss within the conductor gives rise to a compensating inward flow. 

These theoretical patterns are modified in practice by the damping 
due to Joule heat and radiation losses. As a consequence, the successive 



SEC. 8.7] 



LINEAR OSCILLATOR 



443 



current loops diminish in amplitude as one proceeds along the antenna 
outward from the point of feeding. The radiation lobes in or near the 
equatorial plane 6 = ir/2 are strengthened at the expense of the outer 
ones. This effect becomes more pronounced as the resistance of the 
antenna wire is increased. It is illustrated by Fig. 75. The curves were 
obtained by Bergmann 1 from measurements on linear oscillators excited 
in the mode m = 3 and constructed of bronze and two sizes of steel wire 
respectively. The ratio of the current maximum at the central loop to 
that at either of the outer loops was of the order of 1.4:1. When the 




Bronze, 4.0 mm diameter 

Steel, 4.0 mm diameter 

Steel, 0.5 mm diameter 

FIG. 75. Effect of antenna resistance on the radiation according to measurements by 

Bergmann. 

current amplitudes were weighted in this ratio, a calculation of the 
radiation gave results in close agreement with the observed values. 

The power dissipated in radiation is obtained by integration of S* 
over the surface of a sphere of very large radius R drawn about the center 
0. When m is odd, we have 



(16) W = 




S*R Z sin 6 d6 d<t> 



J* 2 (* 

cos 2 I -~- 

sm 



* 

-~- cos 



To evaluate this integral, we replace the variable 6 by u = cos 6 between 
the limits +1 and 1. Then, since 



(16) transforms to 

(18) W = 15/ 3 

1 BERGMANN, Ann. Physik, 82, 504, 1927. 



l-^-ad + u 1 1-u) 

r 1 l + cosm,m du 
J-i 1 + w 



444 RADIATION 

Now let 1 + u = v/mTT, which leads to 



[CHAP. VIII 



TIT 1 C 72 f 1 COS , 

W = 15/g I cfo, 

Jo v 



(19) 
or 

(20) TF = 15J 2 [ln 2miry - Ci 2wwr], 

where 7 = 1.7811 and Ci a; is the cosine integral 

cos v 



(21) 



Ci x = 



(to, 



for which values \iave been tabulated in Jahnke-Emde, "Tables of 
Functions/' 

An identical result is obtained when m is even. In either case the 
radiation resistance is 

(22) (R = 30(ln 2mwy - Ci 2>mr), (m = 1, 2, 3, ), 

or, since In 2iry = 2.4151, 



(23) 



<R = 72.45 + 30 In m - 30Ci 2jrm 
160 



ohms. 



140 

I 12 

o 

i 100 

o 

c 

^3 
| 80 

I " 

""d 

40 
20 



1 2 3 4 5 6 7 8 9 10 11 12 

2/ 
m = ~ , order of excitation 

A 

FIQ. 76. Radiation resistance of a linear antenna at resonance. 

If m > 2, the term in Ci 2irm is negligible. The relation of radiation 
resistance to the order of excitation is shown graphically in Fig. 76. 
Equation (23) is valid only at resonance, when the quantity 2l/\ = m is 
an integer. At intermediate wave lengths the radiation resistance 
follows a curve which oscillates above and below the resonance values 
indicated by Fig. 76. The exact form of this curve depends on the 
nature of the excitation. 1 

1 See, for example, an interesting paper by Labus, Hochfrequenztech. u. Elektroak., 
41, 17, 1933. 



SEC. 8.8] RADIATION DUE TO TRAVELING WAVES 445 

8.8. Radiation Due to Traveling Waves. The current distribution 
considered in the preceding paragraph constitutes a standing wave with 
nodes located at the end points. By a proper termination of the wire, 
a part or all of the reflected wave may be suppressed with the result that 
the current is propagated along the conductor as a traveling wave. 
Such, for example, is the case of a transmission line, where in order to 
secure the most efficient transfer of energy every effort is made to avoid 
standing waves. 

To calculate the radiation losses of a single-wire antenna of finite 
length 1 with no reflected current component, let us assume for the current 
distribution the function 
(24) u(& 



If, as in the previous case, the antenna is fed at the central point, the 
radiation field intensities are 

i 
(25) E^^H^-^I, ^ j**-** J 3 , e-~. m d{ , 

which, when evaluated, gives 



sin -g 1 - cos * 
(26) Ei = -t'60/o sin 9 e 



- - 
1 - cos 6 R 

The radiant energy flow is determined by the function 

kl 
2 sn sn 

/O7\ 

(27) 



~\ 

sin2 e sin2 ho (1 - cos 0) 

J JL 

_ 



_ ___ - . i _ cQg g) - _. 

The power dissipated in radiation is, therefore, equal to 






J*sin 2 0sin 2 [! Z (l - cos 0)1 
- (T^i - - Jsi 



To integrate we put u = 1 cos 6, du = sin 6 d6, and reduce (28) to 

*2 7 7 



(29) 



/' 

Jo 



Omitting the elementary intervening steps, one finds for W the expression 
/ . 4 

I 97 A. 7 Sm "\ 

(30) W - 30/ 2 1.415 + In ^ - Ci ~ + ~ ri A 

\ XX 4?n 

\ T 

1 The problem of a line with parallel return was first treated correctly by 
Manneback, /. Am. Inst. Elec. Engrs., 42, 95, 1923. 



446 



RADIATION 



[CHAP. VIII 



and for the radiation resistance 



sm 



(31) 



<R = 84.9 + 601n ^ 



- 60Ci 

A 



It appears that for equal wave lengths and the same antenna the radiation 
resistance of a traveling wave is greater than that of a standing wave. 



A 

2 




Marconi-Franklin antennas Idealized current distribution 

FIQ. 77. 

8.9. Suppression of Alternate Phases. 

The directional properties of a single- 
wire antenna can be accentuated by 
suppression of alternate current loops. 
This is accomplished in the Marconi- 
Franklin type antenna by proper loading 
with inductance at half -wave-length in- 
tervals along the line. The alternate sec- 
tions of opposite phase are shortened to 
such an extent as to render them negligible 
as radiators and the antenna is then equiv- 
alent to a colinear set of half-wave oscil- 
lators placed end to end and all in phase. 
This is shown schematically in Fig. 77. 
The contribution of each oscillating segment to the radiation field at 
a distant point arrives with a phase advance of ft = ud/c over its next 




FIG. 78. Evaluation 



the series 



; b e-* 



m-0 



SEC. 8.9] SUPPRESSION OF ALTERNATE PHASES 

lower neighbor. Consequently the resultant field is 



447 



cos 



(32) 



t = -6(h7 



(5 



sin 6 



R 



tn-0 



where n is the number of current loops. Now the quantity 6 e~ imft is a 
vector in a complex plane rotated through an angle mf) from the real 
axis. The sum indicated in (32) can, therefore, be evaluated by simple 
geometry. The vector diagram is drawn in Fig. 78, and the sum is 
clearly the chord of a regular polygon. Following the notation of the 
figure we have 



b = 2a sin > 



= 2a sin -~j 

l 



whence V = 



Q 

at an angle a = (n 1) = with the real axis. 



Sin 2 



Since in the present instance 6 = 1, we obtain 
(33) 



. n/3 

8111 IT t(n 1)- 

e-W = -. e 2 . 



sm 2 



For we write o>d/c = TT cos 0, since d = ^ cos ^> an( i so obtain 



cos ( K cos ) sin ( -^- cos 
(34) E e = 60tI - 



sin 



in(^cosg) e ^^- <( n-i)= 
/* \ B 



cos 



sin 



(35) 



in f - cos 0j 
The radiation intensity is 

cos ( ^ cos 6 } sin ( -^ cos J 



5* = 



30 



sin 



(* 

Sin I ;r 





Sin I r COS 



w 



and the total power radiated 

* T cos 2 ( 7: cos 6 } sin 2 



(36) 



F = 30/ 2 



cos 0J 



sin 6 



^d6. 



sin 2 ( cos 



448 



RADIATION 



[CHAP. VII [ 



This integral has been evaluated by Bontsch-Bruewitsch, 1 who obtains 

[nr (n-l)Tr (n-2)ir (n-3)*- rl 

S - 4S + 8S - 125 + - 4(n - 1)S L 
0000 Oj 



where 

mv C m * gm 2 u I 

(38) S = I du = K (In 2W7T7 Ci 2m7r). 

v ' o Jo u 2 

The radiation resistance is, therefore, 

(39) <R = (-] 



nir (n-l)ir (n 2) 

60(5 - 4S + 8S 

000 



600 r 



Figure 79a shows the concentration of radiant energy in the equatorial 

plane for the case of three oscillat- 
ing segments; Fig. 796 is a plot of 
radiation resistance which shows 
that a marked increase of radiating 
efficiency is achieved by the sup- 
pression of alternate interfering 
phases. 

8.10. Directional Arrays. The 
single-wire antenna is directive 
only in the sense that the radiation 
is concentrated in cones of revo- 
lution about the antenna as an axis. 
A colinear set of half-wave oscil- 
lators excited in phase confines the 
radiant energy to a thin, circular 
disklike region such as that of Fig. 
79a. If now one wishes to intro- 
duce a preferred direction in the 
equatorial plane itself, or any other 
form of axial asymmetry, the 
single-wire radiator must be re- 
placed by a group or array of 
half-wave antennas suitably spaced and excited in the proper phase- 
In practice the spacing between centers is usually regular, so that 
the array constitutes a lattice structure. There is in fact a certain 
parallel between the problems of x-ray crystallography and the design 
of short-wave antenna systems. From the x-ray diffraction pattern the 
physicist endeavors to locate the centers of the diffracting atomic dipoles 
and hence determine the structure of the crystal; the radio engineer must 
choose the lattice spacing and phases of the current dipoles so as to 
1 BONTSCH-BKTJEWITSCH, Ann. Physik, 81, 437, 1926. 



500 


? 




400 


- 






300 


- 








200 


- 










100 



_L 










12345 
n 

FIG. 796. 



FIG. 79a. 

FIG. 79a. Radiation pattern about an- 
tenna whose current distribution is indicated 
by the dotted curve. 

FIG. 796. Radiation resistance of n half- 
wave segments oscillating in phase. 



SBC. 8.10] 



DIRECTIONAL ARRAYS 



449 



obtain a prescribed radiation pattern. In both cases the distribution of 
radiation is characterized by a phase or form factor dependent upon the 
polar angles 6 and <, and the methods which have been developed for 
the analysis of crystal structure can be applied in large part to antenna 
design. 

To simplify matters we shall assume that all radiators of the array 
are parallel. The antennas are excited at the central point by means of 




FIG. 80. A system of half-wave oscillators arrayed in a regular lattice structure denned by 

the base vectors ai, fta, &? 

transmission lines which can be designed to radiate a negligible amount of 
energy. Moreover the phase velocity of propagation along these lines 
can be adjusted to values either less or greater than c; consequently, the 
phase relations between the members of the array can be prescribed 
practically at will. 1 

The phase factor of a single half-wave antenna will be denoted by 



(40) 



COS ( 2 COS 1 



sin 



The centers of the oscillators are spaced regularly in a lattice structure 
whose constants are fixed by the three "base vectors " ai, a z , as, as shown 

1 The technical aspects of this problem have been discussed by many recent 
writers. One may consult: Carter, Hansell and Lindenblad, Proc. Inst. Radio 
Engrs.y 19, 1773, 1931; Wilmotte and McPetrie, J. Inst. EUc. Eng. (London), 66, 949, 
1928; Hund, "Phenomena in High-frequency Systems," McGraw-Hill, 1936; Holl- 
man, "Physik und Technik der ultrakurzen Wellen," Vol. II, Springer, 1936. 



450 RADIATION [CHAP. VIII 

in Fig. 80. The jth oscillator is located in the network by a vector 
(41) r, = jiai + J 2 a 2 + jsa 8 , 



where ji, J 2 , js are three whole numbers not excluding zero. The phase 
of the jth oscillator with Tespect to the phase of the oscillator at the 
origin is /?/, 



(42) pj = jil + J2<*2 + J33. 

Thus ai is the phase of any oscillator relative to its nearest neighbor on 
the axis denned by the base vector ai, and so on for the others. The 
radiation field of the jth oscillator is, therefore, 

pikR iut t/sRo-r, 1/3/ 

(43) fy = -t60I #Fo(0) - - ^ - 

where R is a unit vector in the direction of the radius vector R. The 
resultant field intensity is obtained by summing over the entire array. 

pikR i 

(44) E = -t6QF 



By a proper adjustment of amplitude, spacing, and phase in (44) a 
radiation pattern of almost any desired form can be obtained. We shall 
consider only the most common arrangement in which all current ampli- 
tudes are the same and the base vectors ai, a 2 , as define a rectangular 
structure whose sides are parallel to the o>, t/-, and z-axes. In each row 
parallel to the rr-axis there are ni radiators; in each row parallel to the 
2/-axis there are n 2 ; and in each column parallel to the z-axis there are n 3 , 
so that the lattice is completely filled. 

(45) Ro TJ = jiai sin cos <t> + J 2 a 2 sin sin $ + J 8 a 3 cos 6 



in which ai, a 2 , a 3 are the spacings along the x-, y- y z-axes respectively. 
The field intensity at any point in the radiation field is 



e" 



(46) E= -ieo/oFo/i/i/i^-jj , 

whose complex phase factors are 



i sin 8 cos 0+ai) 



n 
(47) fa = V 6 -tf:(fcaj8in fl sin 



SEC. 8.10] 
If we let 
(48) Y i = 



DIRECTIONAL ARRAYS 



451 



sin 6 cos <t> + 
73 = 
then (47) can be written 



72 = ka z sin sin < + 
cos 6 + 3 , 



(49) 



^^ -a, 



'BIO ,y 

T" -^-"r 



sm 



1,2,3) 

' = 1} > 



by (33). The complex factors are eliminated from the radiation intensity 
and we get 

(50) 8 * = &i* 

where 



sin 



(51) 



F.= 



sing 



(s = 1, 2, 3). 



All the radiation formulas derived above are special cases of this 
result. For example, the single wire of Sec. 8.7 excited in an upper mode 
constitutes a linear distribution of half-wave oscillators with ai= a 2 = 0, 
a 3 = X/2, 0:3 = TT. Then 



sm 



(52) 



77T -I Tjl 

i r 2 1, -r 3 



(cos e + 1) 



COS 



which leads directly to (11) on page 441. If the colinear oscillators are 
in phase, as described in Sec. 8.9, we take ai = a 2 = 0, as = X/2, a 3 = 0, 
and obtain (35). 

Consider next a row of vertical half -wave oscillators spaced along the 
horizontal x-axis. We shall assume them to be excited in phase and 
spaced a half wave length apart. a\ = X/2, a 2 = a 3 = 0, ai = 0. Then 



(53) 



30 II 





COS 



( ^ cos 6 J sin ( ~- sin 6 cos $ J 
-~j~~ s - 

sin I ^ sin 6 cos <f> \ 



sin 



which now depends on the equatorial angle <t>. The zeros occur at those 
angles for which the quantity ni7r/2 sin 6 cos <t> has any one of the values 
IT, 2r 9 . . . . In the equatorial plane 8 = ir/2 and there are radiation 



452 RADIATION 

nodes along the lines determined by 
(54) 



ICHAF. VIII 



,2,4 .fti 

cos <t> = + j j 

fti fti fti 



or 



cos 



HI n\ 





Fio. 816. Radiation 
pattern in the equatorial 
plane of four half-wave 
oscillators operating in 
phase and with half-wave- 
length separation. 



FIG. 8 la. Radi- 
ation pattern in the 
equatorial plane of 
two half-wave oscil- 
lators operating in 
phase and separated 
by a half wave length. 

as fti is even or odd. The principal maximum occurs at < = x/2, so that 
this array gives rise to a broadside emission of energy. Subsidiary 
maxima are determined by the condition dFi/d$ = 0, which in the 
present instance leads to the relation 



(55) 



tan 



^ sin 8 cos <t>] = tan f-~- sin 6 cos <l. 



When 6 = <t> = ir/2, FoFi = fti, so that the radiation intensity in the 
direction of the principal maximum is n\ times the maximum intensity of a 
single oscillator. On the other hand, each oscillator receives only the 
nith part of the total power delivered by the source (assuming equiparti- 
tion), and hence the net gain in radiation intensity in the preferred direction 
is given by the factor n\, the number of oscillators. 



SBC. 8.10] 



DIRECTIONAL ARRAYS 



453 



The neighboring zeros lying on either side of the principal maximum 
in the equatorial plane are fixed by the relation cos < = 2/n\. Hence 
the larger m, the narrower the beam. The beam is evidently confined to 
a region bounded by the angles <t>i and <#> 2 , where cos fa cos <t>\ = 4/ni, 




i 

FIG. 82. Radiation pattern in the equatorial plane of two half-wave oscillators 90 deg. out 
of phase and with quarter-wave spacing. 

lying on either side of <t> = ir/2. The radiation patterns in an equatorial 
plane have been plotted in Figs. 81a and 816 for arrays containing two 
and four oscillators. 

Lastly, we shall apply (50) to two antennas spaced a quarter wave 



length apart and 90 deg. out of phase. 
ai = -7T/2. 



Let ai = A/4, a 2 = a s = 0, 



<- ^ 



(56) 



0=77/2 



R 2 



sin 



COS 




3s f j (sin 6 cos <j> 1) J 

The radiation pattern in the equa- "U 
torial plane is shown in Fig. 82. ~ 
The energy is thrown forward in a 
broad beam in the direction of the 
positive x-axis. Radiation to the 
rear is reduced to a very small 
amount. In many commercial in- 
stallations the rearward antenna is not connected directly to the source. 
Its excitation is then due to inductive coupling with the forward oscillator 
and its action is that of a reflector. In this case, however, the spacing is 
not quite A/4 and a small adjustment must also be made in the length of 
the reflector. 



FIG. 83. Radiation patterns in the 
equatorial plane for a horizontal row of 
vertical half-wave oscillators backed by a 
row of reflectors, illustrating the concen- 
tration of radiation in a forward beam by 
increasing the number of oscillators. 
(Hotlmann.) 



454 



RADIATION 



[CHAP. VIII 




FIQ. 84. Radiation patterns in the 
meridian plane for a vertical set of half- 
wave oscillators backed by reflectors. 
(Hollmann.) 



A needlelike beam in space can be obtained by a combination of the 
simple arrays described above. Concentration of the radiation in a 
meridian plane is accomplished by means of a horizontal row of vertical 
oscillators. Extending the length of these vertical lines into a series 

of half-wave radiators excited in 
phase results in a concentration in 
the equatorial plane. Finally, this 
two-dimensional array is backed up 
by a corresponding set of reflectors 
at an approximate distance of X/4. 
In Fig. 83 the radiation diagram in 
the plane = ir/2 of a horizontal 
row of vertical oscillators backed by 
reflectors is drawn for several values 
of n 2 , the number in a row. In Fig. 
84 the same diagram is drawn for a 
vertical set in the meridian plane <t> = ir/2. In both cases the effective- 
ness of an increase in n is apparent. The greatest improvement naturally 
occurs when n is small, in which case the addition of another oscillator 
makes a very considerable difference. The relative gain decreases with 
increasing numbers and a point is soon reached at which any further 
addition to the array is economically un- 
warranted. 

8.11. Exact Calculation of the Field 
of a Linear Oscillator. The method of 
Poynting, which has thus far been em- 
ployed to calculate the radiation resist- 
ance, is based entirely on a determination 
of the outward flow of energy at great 
distances from the center of the radiating 
system. Now if energy is constantly lost 
from the system it must also be supplied 
by the source; consequently, work must 
be done on the antenna currents at a constant rate. There must be a com- 
ponent of the e.m.f . parallel to the wire at its surface which is 180 deg. out 
of phase with the antenna current. A knowledge of the phase relations 
between current and e.m.f. along the conductor must lead to the same value 
of radiation resistance. In fact it can give us more; for this resistance is 
but the real part of a complex radiation impedance which must eventually 
be determined if the radiator is to be treated as a circuit element. 

It is apparent that the results of the preceding paragraphs cannot be 
applied to this alternative method, for they are valid only at great dis- 
tances from the source. It is essential that we have an exact expression 




FIG. 85. Coordinates of the linear 
oscillator discussed in Sec. 8.11. 



SBC. 8.11] EXACT CALCULATION OF THE FIELD 455 

for the field in the immediate neighborhood of each conductor. The 
demonstrations by Brillouin, 1 Pistalkors, 2 and Bechmann 3 that an 
exact expression in closed form can in fact be found were an extremely 
important contribution to antenna theory. 

We shall consider again a single linear conductor coinciding with a 
section of the 2-axis, but the center of the conductor need no longer be 
located at the origin. A point on the conductor is fixed by its coordinate 
, and its length is I = & 1, as shown in Fig. 85. The current dis- 
tribution is sinusoidal, but the current need not be zero at the ends if 
terminal loading is taken into account. Such loading can be accomplished 
through the capacitative effect of metal plates, by grounding through a 
lumped inductance or resistance, or by any of a variety of other devices. 
The current is, therefore, again 
(57) / = 7 sin (fc cfte-*** 

and the Hertz vector of the distribution is 



(58) n.(z, = j~ e-** sin (fc - a) df, 

which can be written also in the form 

r /* * *z. / _i_ t\ /* ! / >\ 

(W\ TT (r t\ 6 P- I - . fit - P ia I e% , 

IcJJ/l JLlV* / (/I ~~~ ,-* I C/ I ~~ \Jd C t/ I " I 

OTTCoO) L J*i r Jfi r 



The cylindrical coordinates of an arbitrary point in the field are 
p, 0, 2, and the distance from any current element to this point is 

r = vV + (z - ) 2 - Now let 

(60) u = r + f - 2, du = - d. 

Then 

(6i) r ^^ dt = ** r d^ 

' Jii r s Jut u 

where Ui = ri + f i 2, w 2 = r 2 + ^2 2?, and n, r 2 are the distances 
from the ends of the wire to the point of observation. Likewise, if 

(62) v = r { + 2, dv = - df, 
\ / s i > r s, 

(63) I ^-^d= --** I ^dw, 
Jfi r Ji y 

where Vi = ri f i + 2, v 2 = r 2 f 2 + 2. The Hertz vector is, there- 

' r /* wa 'fc ^* rs t*jfc ~i 

H, = f e i(kz-a) I ^du + 6 -'<*-> dw 

87T CO L Jui W Ji V J 



fore, 



1 BRILLOUIN, RadiotlectriciU, April, 1922. 

1 PISTALKORS, Proc. Inst. Radio Engrs., 17, 662, 1929. 

* BECHMANN, ibid., 19, 461, 1471, 1931. 



456 RADIATION [CHAP. VIII 

A simple transformation reduces these exponential integrals to a form 
for which tabulated values are available. We note first that 

(* u * e iku C giku C e \ku 

(65) du = du - du. 

1 ' Jui U Jui U Ju, U 

By definition the cosine and sine integrals are 

_.. , f * cos ku , a . , f M1 sin fcw , 

(66) Cikui = - I du, Sifct/i = I au; 

Jui ^ JO W 

since Si < = Tr/2, we have 

/* 00 "J. ' 

(67) I du = Ci kui i Si fc^i + ^> 

Jui ^6 ^ 

or 

/*Ul -L Wl Ml 

(68) I - du = Ci ku + i Si ku 

Jui M Ul Ml 

Tables of these functions will be found in Jahnke-Emde. 
To calculate the field we apply the formulas 

(69) E = vv n + fc 2 n, H - -tcoev x n, 

which in cylindrical coordinates reduce to 



V ' f)IT 

H, = H p = 0, H* - zcoe ^-*- 

Since MI, M 2 , etc. in (64) depend upon both z and p, the integrals must be 
differentiated with respect to their limits. For example, 



_ 
dz ui dz 



Similarly, 



, 

3 

* J. 



ri - (z - f i) ri 

After differentiating in this manner and combining terms one obtains 
for the components of field intensity 

* r^-iw* T (> ikr * e ikri 

(73) E 9 = - 4^ - U cos (fcfc - ) - * - cos (fc n - a) 

N r 



SEO. 8.12) RADIATION RESISTANCE 457 

^7<n tf tToe-*' (k(z - 2 ) e^' ,_ . , 
(74) #, = - <-i - 2_d _ cos (& _ a ) 

co ( p r 2 



cos fcfe -)-- cos 
P 



1* /\P~~ tto * F />*& 

(75) ff, = H - 
w L P 

-L- *(* " &) ^* r ' . fltt . J(Z - fQ eri "I 

_| --- sm ^^ _ a ) -- v - s_y_ - gm ^.^ _ ^ 

P ' 2 p TI J 

In the case of an isolated wire whose ends are free the current will be 
zero at the points = f 1? J = 2 . Then jfe = Trw/Z, where m is again the 
number of half wave lengths along the wire. Since 



fi \ . /7T7tt 2 \ 
sm -j^- - a } = sm ( ^ - a j = 0, 

we place a = "^ = Tm M - ^\ The field intensities then reduce 
to the comparatively simple expressions 

I /jiArs f>\kr\. \ 

E z = -iSO/oc-^ (-l) m ^ -- ~r- h 
(76) iJ A 



8.12. Radiation Resistance by the E.M.F. Method. The expressions 
for the field derived in the preceding paragraph are valid and exact at 
all points of space, including the immediate neighborhood of the current. 
The occurrence of singularities at the end points n = 0, r 2 = 0, results 
from the assumption of a line distribution of current. In practice the 
finite cross section of the wire ensures finite values of the field intensity, 
although relatively large intensities must be expected near these ter- 
minals. At any other point along the conductor the phase relation 
between current and the parallel component of E can be determined from 
(73) or (76) ; consequently, the work necessary to maintain the current 
against the reactive forces of the field can be calculated. 

In Sec. 2.19 it was shown that for real vectors E, H, and J and a cur- 
rent distribution in free space the divergence of the Poynting vector 
satisfies the relation 

(77} v.S=-E.J- 



458 RADIATION [CHAP. VIII 

If the field is periodic, the average of the time rate of change of stored 
energy is zero, so that 

(78) V^S = V S = - E~TJ. 

If now in place of real vectors we deal with complex field intensities and 
currents, we have by Sec. 2.20 



(79) S - $Re(E X H) = 

ReE Re] = $Re(E J), 

the sign ~ indicating conjugate values. The total average power radiated 
from a system of currents is, therefore, 



(80) W 



= I S n da = ~ Re I E J dv. 
Js * Jv 



The radiation can be calculated either by integrating the normal component 
of the Poynting vector over a closed surface including all the sources, or by 
integrating the power expended per unit volume over the current distribution. 
Suppose next that there are n linear conductors carrying current in 
otherwise empty space. Without any great loss of generality it may be 
assumed that the conductors of the system are parallel to one another 
and to the z-axis. The current in the jth conductor is I/, a complex 
quantity. The z-component of the electric field at the jth conductor 
due to the ith current is EH Then by (80) the total radiation from the 
system is 

(81) W = ~ Re 

7^1 

the integrations to be extended the length of each conductor. This is the 
rate at which the induced e.m.f.'s do work on the currents. Let us write 

(82) Jy = Joy sin (fcy ay) e ""*"*, Eji = lQ l Ujie~ l " t . 

The amplitudes Joy and J * are in general complex, and the function 
[/, is determined by (73). Then 

1 n ?^ 

(83) W = \Re 2 2 /( "^ Z *> 

J = l l-l 

where 

(84) ZH = - P Va sin (fefo - ay) df /. 

Jtii 

The quantities Z^ are coupling coefficients or, in a certain sense, transfer 
impedances. The real part of the coefficient Z t < represents the radiation 
resistance of the tth conductor when all other conductors are removed. 



SBC. 8.12] RADIATION RESISTANCE 459 

A very neat method of calculating the radiation has been suggested 
by Bechmann. 1 If the current distribution is sinusoidal as in (82), one 
may write 

d 2 /- 

(85) ^ = -*/,. 

a $i 

The field intensity E,i satisfies the relation 

(86) 
Moreover, 

(87) I < - n i + - j i - n * 

(87) /,___!!, -^ + ^7, n,- 

and hence 

From (64) and (68) we have 

T .p-io>t r r 

(89) n< = ^ - L*,-) Cifcu + 

57TCOW L L 

+ 6 



-.w,-) I Ci kv + i Si kv | 



where, in the present case w = r^ + {, fy, t; = r/ f + f,. The 
coordinate ^i refers to points on the antenna whose current is /, while 
/ is a point on the jth conductor. The subscripts 1 and 2 denote as 
before the coordinates of the lower and upper ends respectively of an 
antenna. No further integration is necessary to find the radiation from a 
system of linear conductors. Since the current amplitudes 7 t and J / are 
factors in (88), an integrated expression for the impedance Z 3i can be 
found. 

In the case of an unloaded antenna the current Ij is zero at the ter- 
minals i/ and 2 ,-. Then fci ; - a,- = 0, fc 2 ,- ay = m,^r, and we obtain 



(90) W = Be 



A difficulty arises when this formula is applied to colinear conductors. 
Consider, for example, the radiation from a single linear oscillator as in 
Sec. 8.7. This case is the limit of two parallel wires whose separation 
approaches zero. If we take the terminals of the two conductors to be 
coincident, then 

1 BECHMANN, loc. cit. t p. 1471. 



460 RADIATION [CHAP. VIII 

when f / = 1, UK = + li 1 = 0, u 2i = Z + 2 1 = 2Z, 

VH = - 1 + 1 = 0, t>2* = I - { 2 + f i = 0, 

and when , = 2 , MI = Z + 1 - 2 = 0, i* 2 < = + 2 - 2 = 0, 

I>K = I - f i + fc = 2Z, u 2 = - 2 + f 2 = 0. 

By (89), 

-fcrf 

Ci kv + * Si kv 



07T CO 



^- r i ws=21 

~ - Cifcu + iSifcw 

OTTCoCO L Jw<=0 



But A; = Trm/l = 2?r/X, VMO/CO = 60; hence by (90) 

/7T 



Tf = ~ 



This result fails, however, since Ci is infinite. The difficulty can be 
avoided by calculating first the radiation of two parallel wires separated 
by a small but finite distance p. Upon passing to the limit p = 0, the 
infinite terms drop out 1 and one obtains, as in Sec. 8.7, 



(91) W = 15/ 2 o(ln 

THE KIRCHHOFF-HUYGENS PRINCIPLE 

8.13. Scalar Wave Functions. Let V be any region within a homo- 
geneous, isotropic medium bounded by a closed, regular surface S, and 
let \I/(x, y, z) be any solution of 



(1) W + k V = 

which is continuous and has continuous first derivatives within V and on 
S. Then it follows directly from Green's theorem, (7), page 165, that 
the value of \l/ at any interior point x r , y f , z' can be expressed as an integral 
of \// and its normal derivative over S. 



where as usual 

(3) r = V(x f ~ *) 2 +~(7/ - yY + (z f - zY 

is the distance from the variable point x, y, z on S to the fixed interior 
point x', y', z'. Equation (2) is in fact the special case of (22), page 427, 
for which all sources are located outside S and the time factor exp ( ivl) 
has been split off. 

1 Details are given by Bechmann, loc. cit. t p. 1477. 



SEC. 8.13] 



SCALAR WAVE FUNCTIONS 



461 



The function defined by (2) is continuous and has continuous deriva- 
tives at all interior points, but exhibits discontinuities as the point 
z f > y'> %' traverses the surface S. The transition of the function 



(*) 



across S was shown to be continuous in Sec. 3.15; consequently, the dis- 
continuities of (2) are identical with those discussed in Sec. 3.17 in 
connection with the static potential. If x', y' ', z' is any point, either 
interior or exterior, the function defined by the integral 



fr\ 
(5) 




FIG. 86. The surface Si may bo closed at infinity. 

is at all interior points identical with the solution ^ whose values have 
been specified over S, and at all exterior points is zero. 1 

As in the static case, too, the values of \j/ and its derivatives at all 
interior points are uniquely determined by the values of ^ alone on S 
(Dirichlet problem), or by d\l//dn alone (Neumann problem). The values 
of both ty and d\l//dn, therefore, cannot be specified arbitrarily over S if 
u and \[/ are to be identical at interior points. The function u defined by 
(5) satisfies (1) and is regular within V whatever the choice of ^ and 
d\l//dn over S, but the values assumed by u and du/dn on S will in general 
differ from those assigned to \l/ and d\l//dn. 

All elements of the surface S which are infinitely remote from both 
the source and the point of observation contribute nothing to the value 

1 The analytic properties of the Kirchhoff wave functions are discussed in detail 
by Poincare", "Th&me mathe'matique de la lumiere," Vol. II, Chap. VII. See also 
the very useful treatise by Pockels, "Uber die partielle Differentialgleichung 
Aw -f k*u = 0," Teubner, 1891, and "Lectures on Cauchy's Problem" by Hadamard, 
Yale University Press, 1923. 



462 RADIATION [CHAP. VIII 

of the integral (2). In Fig. 86 all elements of the source are located 
within a finite distance R\ of a fixed origin Q. The surface S is repre- 
sented as an infinite surface Si closed by a spherical surface S 2 of very 
large radius r about P as a center. The point of observation P(x', t/', z') 
lies within S and all elements of the source are exterior to S. If R, the 
distance from Q to an element of area on 82, is very much larger than 
the largest value of RI, then by (87), page 414, and (32), page 406, 



(6) ~ =* ~ 

n 



n=0 
QO 

(7) TT-l 
hence, 

(8) * = ^ ~ 
Over S 2 



(9) 



The terms in r" 1 , r~ 2 , and r"~ 3 in the expanded integrands of (2), therefore, 
cancel one another, while the remaining terms are of the order r~ n , 
n > 3, and consequently contribute nothing to the integral over S 2 
as r > oo . 

Let us suppose now that the surface S represents an opaque screen 
separating the source from the observer. In virtue of the theorem just 
proved it can be assumed that open surfaces of infinite extent are closed 
at infinity. If a small opening Si is made in the screen, the field will 
penetrate to some extent into the region occupied by the observer. The 
problem is to determine the intensity and distribution of this diffracted 
field. Clearly if the values of ^ and d\f//dn were known over the opening 
Si and over the observer's side of the screen, the diffracted field could 
be calculated by evaluating (2). These values are not known, but to 
obtain an approximate solution one may assume tentatively with Kirch- 
hoff that: 

(a) On the inner surface of the screen 



SEC. 8.13] SCALAR WAVE FUNCTIONS 46S 

(6) Over the surface S\ of the slit or opening the field is identical with 
that of the unperturbed incident wave. 

Nearly all calculations of the diffraction patterns of slits and gratings 
made since KirchhofFs time have been based on these assumptions. 1 
It is easy to point out fundamental analytic errors involved in this 
procedure. In the first place, the assumption that \f/ and d\[//dn are zero 
over the inside of the screen implies a discontinuity about the contour Ci 
which bounds the opening Si, whereas Green's theorem is valid only for 
functions which are continuous everywhere on the complete surface S. 
This difficulty cannot be obviated by the simple expedient of replacing 
the contour of discontinuity by a thin region of rapid but continuous 
transition. If ^ and d\(//dn are zero over any finite part of S, they are 
zero at all points of the space enclosed by /S. 2 In the second place, an 
electromagnetic field cannot in general be represented by a single scalar 
wave function. It is characterized by a set of scalar functions which 
represent rectangular components of the electric and magnetic vectors. 
Each of these scalar functions satisfies (1) and its value at an interior 
point x' , y' , z' is, therefore, expressed by (2) in terms of its values over the 
boundary S. But these components at an interior point must not only 
satisfy the wave equation, they must also be solutions of the Maxwell 
field equations. The real problem is not the integration of a wave 
equation, either scalar or vector, but of a simultaneous system of first- 
order vector equations relating the vectors E and H. 

In spite of such objections the classical Kirchhoff theory leads to 
satisfactory solutions of many of the diffraction problems of physical 
optics. This success is due primarily to the fact that the ratio of wave 
length to the largest dimension of the opening in optical problems is 
small. As a consequence the diffracted radiation is thrown largely 
forward in the direction of the incident ray and the assumption of zero 
intensity on the shadow side of the screen is approximately justified. 
Measurements are usually of intensity and do not take account of the 
polarization. As the wave length is increased, the diffraction pattern 
broadens. A computation of ty from (2) now leads to intensities directly 
behind the screen which are by no means zero, contrary to the initial 
assumption. Various writers have suggested that this might be con- 

1 On the classical theory of diffraction see, for example, Planck, "Einfiihrung in 
die theoretische Optik," Chap. IV, Hirzel, Leipzig, 1927, or Born, "Optik," Chap. 
IV, Springer, Berlin, 1933. To obtain a clearer understanding of the physical signif- 
icance as well as the shortcomings of Kirchhoff's method the reader should consult 
the papers by Rubinowicz, Ann. Physik, 63, 257, 1917, and Kottler, ibid., 70, 405, 
1923. Since the writing of this and the following sections a treatment of the subject 
has appeared in book form: "The Mathematical Theory of Huygens' Principle, " by 
Baker and Copson, Oxford, 1939. 

2 A direct consequence of Green's theorem. See POCKELS, loc. cit., p. 212. 



464 RADIATION [CHAP. VIII 

sidered as the first of a series of successive approximations, the values of 
^ and d\f//dn over S obtained from the first calculation to be used as the 
boundary conditions in the second. There is, however, no proof that 
the process converges and the difficulties of integration make it impractical. 

Recent advances in the technique of generating ultrahigh-frequency 
radio waves have stimulated interest in a number of problems previously 
of little practical importance. A natural consequence of this trend 
towards short waves is the application of methods of physical optics to 
the calculation of intensity and distribution of electromagnetic radiation 
from hollow tubes, horns, or small openings in cavity resonators. In 
radio practice, however, the length of the wave is commonly of the same 
order as the dimensions of the opening, and the polarization of the dif- 
fracted radiation is easily observed. It is hardly to be expected that 
the Kirchhoff formula (2) can be relied upon under these circumstances. 

8.14. Direct Integration of the Field Equations. 1 The problem of 
expressing the vectors E and H at an interior point in terms of the 
values of E and H over an enclosing surface has been discussed by a 
number of writers. 2 A simple and direct proof of the desired result can 
be obtained by applying the vector analogue of Green's theorem to the 
field equations. In Sec. 4.14 it was shown that if P and Q are two vec- 
tor functions of position with the proper continuity, then 

(10) 



where S is as usual a regular surface bounding the volume V. 

Let us assume that the field vectors contain the time only as a factor 
exp ( iwt) and write the field equations in the form 

(I) V X E - iw/iH = - J*, (III) V H = - p*, 

(II) V X H + tucE = J, (IV) V E = J p. 

The medium is assumed to be homogeneous and isotropic, and of zero 
conductivity. The quantities J* and p* are fictitious densities of 
"magnetic current 77 and " magnetic charge, 77 which to the best of our 

1 Sections 8.14 and 8.15 were written in collaboration with Dr. L. J. Chu. 

2 LOVE, Phil. Trans., (A) 197, 1901; LARMOR, Lond. Math. Soc. Proc., 1, 1, 1903; 
IGNATOWSKY, Ann. Physik, 23, 875, 1907, and 25, 99, 1908; TONOLO, Annali di Mat., 
3, 17, 1910; MACDONALD, Proc. Lond. Math. Soc., 10, 91, 1911 and Phil. Trans., (A) 
212, 295, 1912; TEDONE, Line. Rendi., (5) 1, 286, 1917; KOTTLER, Ann. Physik, 71, 
457, 1923; SCHELKUNOFF, Bell. System Tech. J., 15, 92, 1936; BAKER and COPSON. 
toe. tit., Chap. III. 



SEC. 8.14] INTEGRATION OF FIELD EQUATIONS 465 

knowledge have no physical existence. However, we shall have occasion 
shortly to assume arbitrary discontinuities in both E and H, discon- 
tinuities which are in fact physically impossible, but which would be 
generated by surface distributions of magnetic current or charge were 
they to exist. Currents and charges of both types are related by the 
equations of continuity, 

(V) V J - ip = 0, V - J* - twp* = 0. 

The vectors E and H satisfy 

(H) V X V X E - fc 2 E = itt/J - V X J*, 

(12) V X V X H - fc 2 H = io) J* + V X J, 

with fc 2 = w V 

In (10) let P = E, Q = <a, where a is a unit vector in an arbitrary 
direction and <t> = e ikr /r. Distance r is measured from the element at 
x, y, z to the point of observation at x', T/', z' and is defined by (3). We 
have identically 

(13) v X Q = V< X a, v x V X Q = afc 2 </> + V(a V<), 

V X V X P = fc 2 E + tco/J - V X J*. 

Following the procedure of Sec. 4.15, it is easily shown that a is a factor 
common to all the terms of (10) and, since a is arbitrary, it follows that 

,. (*/. T 1 \ 

(14) I I teo/ij9 V X J < H pV< I dv 
<J v \ * / 

= I [icoA*(n X H)< + (n X E) X V</> + (n E)V< n X J*<6l da. 
Js 

The identity 

(15) V X ]*<p dv = I n X J*< da + I J* X V< dv 
Jv Js Jv 

reduces this to 

(16) I UCOM!* ~ J* X V<t> + - pV<t>) dv 

= I [iw/i(n X H) + (n X E) X V<t> + (n E)V<] da. 
Js 

The exclusion of the singularity at r = was described in Sec. 4.15. 
A sphere of radius r t is circumscribed about the point a;', t/, z', its normal 
directed out of V and consequently radially toward the center. 

/I \ >&? 

(17) V</> = ( - - ik ) to, 



466 RADIATION [CHAP. VIII 

and on the sphere n = TO- The area of the sphere vanishes with the 
radius as irr\ and, since 

(18) (n X E) X n + (n E)n = E, 

the contribution of the spherical surface to the right-hand side (16) 
reduces to 47rE(x', y'j 2')- The value of E at any interior point of V is, 
therefore, 



(19) E(z', y', z') = - iurft - J* X V* + P V0 dv 



T~ I [W(n X H)< + (n X E) X V<f> + (n E)V<] da. 

An obvious interchange of vectors leads to the corresponding expression 
forH, 



(20) H(x', ?/, z') = ^ to>j** + J X V0 + P * 

+ 4- ( [*wc(n X E)< - (n X H) X V$ - (n H)V</>] da. 

This last is the extension of (23), page 254, to the dynamic field. 

If all currents and charges can be enclosed within a sphere of finite 
radius, the field is regular at infinity and either side of S may be chosen 
as its "interior." 

In the earlier sections of this chapter the calculation of the fields of 
specified distributions of charge was discussed in terms of vector and 
scalar potentials and of Hertzian vectors. We have now shown that 
E and H may be calculated directly without the intervention of poten- 
tials. The surface integrals of (19) and (20) represent the contributions 
of sources located outside S. If S recedes to infinity, it may be assumed 
that these contributions vanish. Discarding the densities of magnetic 
charges and currents, one obtains the useful formulas 



(21) 



Since the current distribution is assumed to be known, the charge density 
can be determined from the equation of continuity. 

In Sec. 9.2 it will be shown that an electromagnetic field within a 
bounded domain is completely determined by specification of the tan- 
gential components of E or H on the surface and the initial distribution 
of the field throughout the enclosed volume. It follows that when 
a X E and n X H in (19) and (20) have been fixed, the choice of n E 



SEC. 8.14] INTEGRATION OF FIELD EQUATIONS 467 

and n H is no longer arbitrary. The selection must be consistent with 
the conditions imposed on a field satisfying Maxwell's equations. The 
same limitations on the choice of ^ and d\l//dn were pointed out in Sec. 
8.13. The dependence of the normal component of E upon the tangential 
component of H is equivalent to that of p upon J. 

Let us suppose for the moment that the charge and current distribu- 
tions in (19) are confined to a thin layer at the surface S. As the depth 
of the layer diminishes, the densities may be increased so that in the 
limit the volume densities are replaced in the usual way by surface 
densities. If the region V contains no charge or current within its 
interior or on its boundary S, the field at an interior point is 



= ^ I [tM 



(22) B(z',y 

It is now clear that this is exactly the field that would be produced by a 
distribution of electric current over 8 with surface density K, a distribu- 
tion of magnetic current of density K*, and a surface electric charge of 
density 77, where 

(23) K - -n X H, K* - n X E, r, = -e n E. 

The values of E and H in (23) are those just inside the surface S. 

The function E(x f , y f , z') defined by (22) is discontinuous across 8. 
It can be shown as above that discontinuities associated with < = e lkr /r 
are identical with those of the stationary regime, $ = 1/r. Then by 
Sec. 3.15 the integral 

(24) Ea(s', 

suffers a discontinuity on transition through S equal to n AE 3 = iy/c, 
where AE 3 is the difference of the values outside and inside. The third 
term of (22), therefore, does not affect the transition of the tangential 
component but reduces the normal component of E to zero. Likewise by 
Sec. 4.13 the discontinuity of 

(25) Et(x' 9 ?/, ') = r- I K* X V< da 

^" *J s 

is specified by n X AE 2 = K*, so that the second term in (22) reduces the 
tangential component of E to zero without affecting the normal com- 
ponent. The first term in (22) is continuous across $, but has discon- 
tinuous derivatives. The curl of E at a; 7 , j/', z f is 



(26) V' X E(x', y', z'} = ~ (n X H) X V* da 

s 




468 RADIATION [CHAP. VIII 

which is of the type (25). The vector E and the tangential component of 
its curl are zero on the positive side of S; E is, therefore, zero at all external 
points. The same analysis applies to H. 

8.15. Discontinuous Surface Distributions. The results of the pre- 
ceding section hold only if the vectors E and H are continuous and have 
continuous first derivatives at all points of S. They cannot, therefore, 
be applied directly to the problem of diffraction at a slit. To obtain 
the required extension of (22) to such cases, consider the closed surface S 
(surfaces closed at infinity are included) to be divided into two zones 

Si and $2 by a closed contour C ly- 
ing on S, as in Fig. 87. The vectors 
E and H and their first derivatives 
are continuous over Si and satisfy 
the field equations. The same is 
true over /S 2 . However, the com- 
ponents of E and H which are tan- 
gential to the surface are now 
FIQ. 87. C is a contour on the closed subject to a discontinuous change 

surface S dividing it into the two parts Si in passing acrogs C from Qne ZQne to 

the other. The occurrence of such 

discontinuities can be reconciled with the field equations only by the 
further assumption of a line distribution of charges or currents about the 
contour (7. This line distribution of sources contributes to the field, 
and only when it is taken into account do the resultant expressions 
for E and H satisfy Maxwell's equations. 

A method of determining a contour distribution consistent with the 
requirements of the problem was proposed by Kottler. 1 It has been 
shown that the field at an interior point is identical with that produced 
by the surface currents and charges specified in (23). A discontinuity 
in the tangential components of E and H in passing on the surface from 
zone Si to zone $2 implies therefore an abrupt change in the surface 
current density. The termination of a line of current, in turn, can be 
accounted for according to the equation of continuity by an accumulation 
of charge on the contour. Let ds be an element of length along the 
contour in the positive direction as determined by the positive normal 
n, Fig. 87. Let n t be a unit vector lying in the surface, normal to both n 
and the contour element ds, and directed into zone (1). The line den- 
sities of electric and magnetic charge will be designated by <r and <r*. 
Then Eqs. (V), when applied to surface currents, become 

(27) m (Ki - K,) = tW, ni - (Kf - K?) = war*, 

1 KOTTLBR, Ann. Phyrik, 71, 457, 1923. 



SEC. 8.15] DISCONTINUOUS SURFACE DISTRIBUTIONS 469 

and hence by (23) 

tW = ni (n x H 2 - n X Hi) = (H 2 - HI) - (ni X n), 
twcr* = ni - (n X E! - n X E 2 ) = -(E 2 - Ei) (ni X n). 

The vector HI X n is in the direction of rfs. If S* represents an opaque 
screen over which E 2 = H 2 = 0, the field at any point on the shadow side 
is 

(29) E(x>, y', *') = -i -L f o V* Hi ds 

- ~ [tw M (n X Hi)* + (n X Ei) X V* + (n . Ei)V<] do, 

*" y $1 

which can be shown to be identical with 

(30) 4E(z', j/', = i- ^ V<#> Hi - ds + <^ Ei X da 

t-CO / C J C 



For the magnetic field one obtains 
(31) 

+ [to>(n X E00 - (n X Hi) X V<^ - (n HO V0] 

*/ <Si 

= 1 ^ v*E 1 -cfa + ^ ^HiXds- f (n^^-^ 
iwjjijc jc Jsi\ dn dn 

It remains to be shown that the fields expressed by these integrals are 
in fact divergenceless and satisfy (I) and (II). Consider first the diver- 
gence of (29) at a point x l ', t/', z r . 



(32) V E(*', y', z') = 



S Is [t 



V<t> + (Q ' El)v2 ^ da 



^ f 

"" */ 01 



taking into account the relation V' = V when applied to < or its 
derivatives. Now 

(33) f (n X H) V< da = f <t>V X H n da - ( <t> H - ds. 

J8 JS J C 



470 RADIATION [CHAP. VIII 

The line integral resulting from this transformation is zero when S is 
closed, but otherwise just cancels the contour integral in (32). Inversely, 
only the presence of the contour integral in (29) leads to a zero divergence and 
waves which are transverse at great distances from the opening Si. From 
(II) and the relation k 2 /iu = icoju, follows immediately the result 

(34) V'-E(o;', 7/, z') =0. 

An identical proof holds for H(a/, ?/', z'). 

Finally, it will be shown that (29) and (31) satisfy (I) and (II). 

(35) V X H(s', y', *') = ~ f [(n X H a ) VV^> + W<t>(n X HO 

47T JSi 

- fw(n X Ei) X V</>] da, 
since the curl of the gradient is identically zero. Furthermore, 

(36) f (n X HO VV0 da = - f ' n (V v ^ X HI) V0 da 

/Ol /01 

= f (n - V X HO V* da - f n X V (HiV0) da 

/<Si ,/$i 

= (p V< Hi ds iut f (n-EOV^da, 

a/ C / Si 

where the operator V V acts on V</> only. The last integral takes account 
of the fact that the field equations are by hypothesis satisfied on /Si. 
Then 

(37) V X H(*', ?/, z') = f c V<t> Hi - ds + J ft UW(n X HO* 

+ (n X EO X V* + (n E 



The validity of (I) is established in the same manner. 1 

FOUR-DIMENSIONAL FORMULATION OF THE RADIATION PROBLEM 

8.16. Integration of the Wave Equation. In Sec. 1.21 it was shown 
how by means of tensors the field equations could be written in an exceed- 
ingly concise form. If the imaginary distance x = id be introduced as a 
coordinate in a four-dimensional manifold, the equations of a variable 
field are in fact formally identical with those which govern the static 
regime, and the methods which were applied to the integration of Poisson's 
equation can be extended directly to the more general case. The four- 

1 These formulas have been applied by L. J. Chu and the author to the calculation 
of diffraction by a rectangular slit in a perfectly conducting screen. The results 
compare favorably with those obtained by Morse and Rubcnstein, Phys. Rev., 54, 895, 
1938, who solved a two-dimensional slit problem rigorously by introducing coordinates 
of a hyperbolic cylinder. Phys. Rev., 56, 99* 1939. See also the treatment of such 
problems by Schelkunoff, ibid.> p. 308, 



SBC. 8.16] INTEGRATION OF THE WAVE EQUATION 471 

dimensional theory is more abstract than the methods described earlier 
in this chapter, and consequently has not been applied to the practical 
problem of calculating antenna radiation. Quite apart from its formal 
elegance, however, the four-dimensional treatment sometimes leads in 
the most direct manner to very useful results. This is particularly true 
of the rather difficult problem of calculating the field of an isolated 
charge moving in an arbitrary way. 

The discussion will be confined to the case of charges and currents in 
free space. As in (82), page 73, the vector and scalar potentials can be 
represented by a single four-vector whose rectangular components are 

n 

(1) $1 = A X) $ 2 = A VJ $3 = A z , $4 = - 0. 

c 

Likewise the current and charge densities are represented by a four-vector 
whose components are 

(2) Ji =* J X) Jz = Jy, t/3 Jz) J 4 = ICp. 

The four-potential satisfies the relations 
(3) 

According to the notation of (35), page 64, D is a symbolic four-vector 
whose components are d/dxk. Then (3) can be expressed concisely as 

(4) D 2 4> = -MoJ, D * = 0. 

Let V be a region of a four-dimensional space bounded by a three- 
dimensional "surface" S. n is the unit outward normal to S. Then in 
four as in three dimensions 



(5) 



I D & dv = I <I> n da. 
Jv Js 



If u and w are two scalar functions of the four coordinates Xi, x*, x 3 , x 
which together with their first and second derivatives are continuous 
throughout V and on S, then 

(6) I D (uHw) dv = I uHw n da, 
Jv Js 

(7) I Hu*nwdv+ I uD*w dv = I u-^-da. 
Jv Jv Js dn 

Thus, one obtains a four-dimensional analogue of Green's theorem, 

(8) r <*D - M D%) * - s g - u fj da. 



472 RADIATION [CHAP. VIII 

Let x'j be the coordinates of a fixed point of observation within S, 
and Xj those of any variable point in V or on S. The distance from 
x 9 - to x^ is 



It can be verified by direct differentiation that the equation 
(10) D 2 w = 

is satisfied by 

(ID - jjj 

at all points exclusive of the singularity R = 0, which will be excised in 
the usual fashion by a sphere Si of radius 12 1. If u is identified with any 
component $, of the four-potential, there follows 



Over Si we have 



To determine the area of the hyperspherical surface S\ polar coordinates 
are introduced, 

Xi = R cos 0i, # 2 = 12 sin 0i cos 2 , 

x 3 = 12 sin 0i sin 2 cos 0, #4 = 12 sin 0i sin 2 sin 0, 

which satisfy 

(15) x\ + x\ + x\ + x\ = 12 2 . 

The scale factors hi are calculated as in (70), page 48, and are found to be 

(16) hi = 1, h* = 12, hz = 12 sin 0i, A 4 = 12 sin 0i sin 2 . 
The element of volume is 

(17) dv = hihjijii dR dOi d0 2 d<t> = 12 3 sin 2 0! sin 2 cZ12 dOi dB 2 d<#>; 
hence, the area of the hypersphere is 

/r pr /2ir 

(18) 12 3 sin 2 0i sin 2 c?0i ^0 2 d<fr = 27r 2 12 3 . 

Jo Jo Jo 

Therefore, 



as 



SEC. 8.17] FIELD OF A MOVING POINT CHARGE 473 



Upon replacing D 2 ^,- by /W/ and recombining the components <!>, into 
a four-vector, one obtains for <I> at the point x'j the expression 



To find the field vectors one must compute the components of the 
tensor 2 F at the point x\. By (79), page 72, 

(21) F * - w, ~ f| * - *> 2 > 3 - *> 

Let us suppose for the moment that all sources are contained within V. 
Then the integral over S contributes nothing in (20) to the value of 
and in this case 



.gf (JXR)**, 

\j y ** 



where (J X R)/* = J/JB* JkRj, and where R is the radius vector drawn 
from x to x'. The cross product J X R is a six-vector or antisymmetric 
tensor whose components are defined as in (62), page 69. In concise 
form, 

(23) *(,>) - 



8.17. Field of a Moving Point Charge. An isolated charge q moves 
with an arbitrary velocity v in free space. It will be assumed that the 
distance from the charge to the observer is such that the charge can be 
represented by a geometrical point. The motion of the charge along its 
trajectory is specified by expressing its coordinates as functions of t, 

(24) xi = /i(0, *2 = /(0, *3 = /s(0, *4 = / 4 (0 = id. 



One must note that here t is not the observer's time but time as measured 
on the charge. Then 

(25) J_ _ J_^ J_^ J f dxi dx, dx 3 = q*ji = q &, (j = 1, 2, 3, 4), 
and by (20) 

< 26 > *- 



The origin on the time axis can be shifted at will without loss of 
generality. It will be convenient to assume that the observer's time 



474 



RADIATION 



[CHAP. VIII 



t' at the instant of observation. Then x( = and 
(27) R 2 = r 2 + xl 

where r is the distance in three-dimensional configuration space from the 

charge to the observer. The poles of the 
integrand in (26) are located at 



-Plane 



4 = r 



(28) 



#4 = ir, 



= +ir. 



Now the data of the problem specify the 
coordinates of the trajectory for real values 
t which are less than t'j and in the present 
instance less, therefore, than zero. Thus 
the fj(t) are given only for values of #4 
on the negative imaginary axis. However, 
tlie assumption will be made that the/,(0 
are analytic functions of z 4 = ict and that 

, , . , . . .. . 

they can be continued analytically over the 
entire complex x 4 -plane. Since the only singularities of (26) are the poles 
at (28), it follows from Jordan's lemma, page 315, that the contour of 
integration can be deformed from the real axis to a small circle about the 
pole #4 = ir in the direction indicated by Fig. 88, and the integral then 
evaluated by the method of residues, page 315. Expansion of R 2 about 
the point x = ir gives 



-100 

FIQ. 88. The contour of inte- 
gration is reduced to a small circle 
about the pole at 4 = ir. 



(29) 

( . 
(30) 
hence, 

, Q1 , 
(31) 



(x, 



2i 



2i 



2i 
= v r 



. .. (v r \ 

2ir = 2ir [ - 1 )> 
\c / 



where v r is the component of charge velocity in the direction of the radius 
vector r drawn from charge to observer. 
The integral (26) now has the form 



(32) 



2r(l- 



- 
dt x^ + ir 



dx<. 



The contour is traced in the clockwise direction; consequently, the 
integral in (32) is by Cauchy's theorem equal to 27ri-~ 

CLL 



SEC. 8:I7J FIELD OF A MOVING POINT CHARGE 475 

(33) *' = ;ra* 



or, in terms of vector and scalar potentials, 

ff)A\ A MO (?V 

(d4) A = 7 ,-, </> == -- 



These are the formulas of Li&iard and Wiechert. The values of r and v 
are those associated with q at an instant preceding the observation by the 
interval r/c. 

Rather than differentiate these expressions to obtain the field vectors, 
it is most direct to apply the same procedure to (23). This has been done 
by Sommerfeld 1 who obtains formulas, the derivation of which by older 
methods led to great complications. It is convenient to resolve the 
expressions for the field into two parts : a velocity field which contains no 
terms involving the acceleration v, and an acceleration field which vanishes 
as v goes to zero. For the velocity field one finds 



(35) ^ 

where 

(36) 7 - 



v<r 



c 
and r is a unit vector in the direction of r from charge to observer at the 

instant t 1 These formulas can be obtained also by applying the 

c 

Lorentz transformation (111), page 79, to the field of a static charge and 
noting that (111) refers to the observer's time. 
The acceleration field is 



(37) 



SOMMERFELD, in Riemann- Weber, "Partiellen Differentialgleichungen der 
mathematischen Physik," p. 786, 8th ed., 1935. See also, ABRAHAM, "Theorie der 
Elektrizitat," Vol. II, pp. 74jf., 5th ed., 1923, and FRENKEL, "Lehrbuch der Elek- 
trodynamik," Vol. I, Chap. VI. 



476 RADIATION [CHAP. VIII 

Note first that both velocity and acceleration fields satisfy the condition 

(38) H 

The magnetic vector is always perpendicular to the radius vector drawn from 
the effective position of the charge, by which one means the position at the 

instant t' - The electric vector of the velocity field is not transverse, 
c 

but in the acceleration field 



(39) E = ^ H X r , E - r = H - r = 0. 

The velocity field decreases as 1/r 2 , while the acceleration field diminishes 
only as 1/r. At great distances the acceleration field predominates; it is 
purely transverse and it alone gives rise to radiation. 

In case the velocity of the charge is very much less than that of light, 
the equations (37) for the field at large distances become approximately 



(4o) 

H~-_-vXr , 

47TTC 

which remind one of (27), page 435, for the radiation field of a dipole. 

The results of this section have been based on the assumption that 
the motion and trajectory of the charge are known, just as earlier the 
assumption was made that the current distribution can be specified. 
In neither case is it exact to treat the problem of the motion or distribu- 
tion apart from that of the radiation. Let us suppose, for example, that 
a charge is projected into a known magnetic field. If the radiation is 
ignored, the mechanical force exerted on the charge is gv X B, from which 
the motion can be calculated by the methods of classical mechanics. 
In case the velocities are large, a correction can also be made for the 
relativistic change in mass. Now the effect of this force is to accelerate 
the charge in a direction transverse to its motion and, consequently, to 
introduce an energy loss through radiation. The dissipation of energy 
through radiation is not accounted for by the force gv X B. An addi- 
tional force, the radiation reaction, which can be compared roughly to 
friction, must be included. The radiation reaction in turn affects the 
trajectory, whence it is obvious that the exact solution can be found only 
by introducing the total effective force from the outset a very much 
more difficult problem. Fortunately the radiation reaction is in most 
cases exceedingly small, so that a satisfactory approximation for the 
motion can be obtained by ignoring it entirely. 



PROBLEMS 477 



Problems 

1. Show that the r.m.s. field intensity at large distances from a half-wave linear 
oscillator is given by the formula 



w cos \i cos 7 



7\/W ' 

E _JL x: -L volts/meter, 

R sin e 

where W is the radiated power in watts, R the distance from the oscillator in meters, 
and 9 the angle made by the radius vector R with the axis of the oscillator. 

Show that the r.m.s. field intensity at large distances from an electric dipole is 

E = 6.7 ~^- \/W volts/meter, 

provided the wave length is very large relative to the length of the dipole. 
2. Let \f/ be a solution of the equation 



, 

which together with its first and second partial derivatives is continuous within and 
on a closed contour C in the xy-pl&ne, and let 



= i 

4jcL 



where r \/(x' #) 2 -f- (y 1 ~ 2/) 2 > ^ denotes the outward normal to C, and 

is a Hankel function. Show that, if the fixed point x', y' lies outside the contour C, 

then u = 0, while u = ^ if it lies within. 

The theorem holds also when H^(kr) is replaced by the function N(kr). This 
is a two-dimensional analogue of the Helmholtz formula (5), page 461, derived by 
Weber, Math. Ann., 1, 1-36, 1869. 

3. Discuss the analogue of the Kirchhoff formula in two dimensions. A proof 
based on the Weber formula of Problem 2 has been given by Volterra, Ada Math., 
18, 161, 1894. (Very interesting work on the propagation of waves in a two-dimen- 
sional space has recently been done by M. Riesz. See the discussion by Baker and 
Copson, "Huygens' Principle," Cambridge University Press, p. 54, 1939.) 

4. Let x', y' be any fixed point within a plane two-dimensional domain bounded 
by a closed curve C, and let ^ be a solution of 



' 
dx* dy* 

which is continuous and has continuous first derivatives on C and within the enclosed 
area S. Show that 

', 2/0 - I K Q (ikr) - * ~ K*(ikr) \ ds + ~ I (*, y) K Q (ikr) da, 
JtC[_dn dn J ** JS 

where r 2 = (x r x) s -f (y f T/)* and K Q (ikr) is a modified Bessel function dis- 
cussed in Problem 10, Chap. VI. The normal n is drawn outward from the contour. 



478 RADIATION [CHAP. VIII 

6. Let the current distribution throughout a volume V be specified by the func- 
tion J(x)e~ tut y where x stands for the three coordinates x, y, z. Show that the electric 
intensity due to the distribution can be represented by the integral 



C 
Jv 



KJ * 



where x' stands for the three coordinates of the point of observation, k = w VM*, and 

r* - (x' - z) 2 + (y' - y) 2 + V - *) 2 . 
6. When the theorem of Problem 5 is applied to a perfect conductor one obtains 



where K is the surface current density at a point x, y, z on the conductor S. Let S 
now be the surface of a linear conductor. Assume that the curvature of the wire is 
continuous and that at all points the cross section is small in comparison with the 
radius of curvature and with the wave length. Show that the field of such a linear 
conductor is given by 



(*',) - ---*"*' 7 

r 



si C dl e lkr ) ic*u C e lkr 
+ ^~<fe>+^ I d,, 

s 2 Jc ds r ) 4^ Jc r 



where V' is applied at the point of observation and the integrals are extended along 
the contour C of the wire between points si and S*. 

7. Apply the formula of Problem 6 to obtain the field of a linear oscillator of 
length I and compare with the results of Sec. 8.11. 

8. In case the linear circuit of Problem 6 is closed, Si = s 2 and the integrated 
term is zero. On the surface of the conductor the tangential component of E is 
approximately 



f^ 2 /" 1 f> lkr 

ii + tv d.-o, 

|_ ds 2 J r 



whence the current / must be of the form 

/ A m e~ lk m' t 



I 



where I is the length of the circuit and m is an integer. Show that the field of such 
a closed, oscillating loop is given by 



The expression is exact in the limit of vanishing cross section and is approximately 
correct if the cross section is small in comparison with the wave length and the radius 
of curvature at each point of the circuit. Its application to Hertzian oscillators, 
such as were used in the early days of wireless telegraphy, was discussed in an Adams 
Prize essay by Macdonald in 1902 entitled "Electric Waves. " 

9. A semi-infinite linear conductor carrying a current of frequency o>/27r coincides 
with the negative z-axis of a coordinate system. Find expressions for the components 



PROBLEMS 479 

of electric and magnetic field intensity and calculate the total energy crossing a plane 
transverse to the conductor. 

10. Compute the radiation resistance of a linear half-wave oscillator by the 
e.m.f. method described in Sec. 8.12. 

11. An isolated, linear, half-wave antenna radiates 50 kw. at a wave length of 
15 meters. Plot the parallel and perpendicular components of electric fiek intensity 
along the antenna on the assumption of a harmonic current distribution and a No. 4 
wire. 

12. The odd and even functions 

8ln (a C S CQS (* CQS 



sin sin 

occur frequently in the theory of linear oscillators. Let TJ = cos 6 and show that 
these function are solutions of 

(1 - T7 2 )F" - 2ijF' + <* 2 (1 - T7 2 ) - - \F = 0. 
L 1 ~ ^J 

Put F = \/l 77 2 G y and show that G satisfies the associated Mathieu equation 
(6), page 375, 

(1 - ^)w" - 2 (a + l)-nw' + (b - cV)w = 

for the particular case a = 1, 6 = a 2 l,c = a. 

Although F (fl) and F<> are periodic in for all values of a, note that they remain 
finite at the poles only for certain characteristic values. For the even function, these 

values belong to the discrete set a% = TT; for the odd function, they form the 

set o mir, where m is an integer. 

Demonstrate the orthogonality of the functions expressed by 

F^F^(1 tf)d-n =0, m 5* n, 

m ? n 

the subscripts referring to characteristic values of , and show that the functions are 
normal. Discuss the relation of these functions to the radiation field of an arbitrary 
current distribution on a linear antenna. 

13. An electromagnetic source is located at a point PI, and another operating 
at the same frequency is located at P 2 . The intervening medium is isotropic but 
not necessarily homogeneous. All relations between field vectors are linear, and the 
time variation is harmonic. Let the field vectors due to the source at PI be EI and 
Hi, those due to the source at P 2 be E 2 and H 2 . Show that wherever these vectors 
are continuous and finite, they satisfy the symmetrical relation 

V (Ei X H 2 - E 2 X HO - 0. 

This result is due to Lorentz and has been developed into a number of reciprocity 
theorems of fundamental importance for radio communication. 



480 RADIATION [CHAP. VIII 

14. On page 434 it was shown that the Hertz vector of an electric quadrupole is 



n = p( 

P 



k(L 4__!_ 

\. V* w 

3 

... ^1 dR C 

Pi = > P.; ' p./ = .,PO *>, 

^T *"' Jr 



where po(i, 2, s) is the density of charge. Let PR, pe, p<t> be the spherical components 
of the vector p (2) . Show that the radiation field of the quadrupole, in the region 
1, is given by 



Ee~* - 



while the radial components vanish as l/R*. Show that the mean radiation inten- 
sity is 



16. Show that the components pe and p^ of the vector p< 2) in the preceding prob- 
lem are related to the components p/ of the quadrupole tensor by 

pe * i sin 20 (pa cos 2 </> + pn sin 2 <#> p 88 -f- pi 2 sin 2<#>) 

-r- cos 29 (pis cos </> + p 2 sin ^), 
p^ J sin (? sin 20 (p 22 pn) + pit sin cos 2< -f ^23 cos cos ^ psi cos sin 0. 

Rotate the coordinate system to coincide with the principal axes of the quadrupole 
tensor. Then pit p M = p 3 i = and the above reduces to 

p^ sin 20 (Qi Q 2 cos 2^), p<t> 2 sin sin 2</> Q 2 , 
where 

Qi - i(pn + ?>22 - 2p8 3 ), Qz = l(pn ~ pii). 



Show now that the total quadrupole radiation is 

W - J- (Ql -f 3$) watts. 

If the medium about the quadrupole is air, /* - 1/c 2 , and \/~ ^ 60, so that 



16. A charge e is located at each end of a line rotating with constant angular 
velocity fc >out a perpendicular axis through its center. A charge 4-2e is fixed at the 
center. ..'he dipole moment of the configuration is zero. Calculate the components 
of the quadrupole moment and find the total radiation. (Van Vleck.) 

17. Two fixed dipoles are located in a plane, their axes parallel but their moments 
directed in opposite sense. The dipoles rotate with constant angular velocity about 



PROBLEMS 481 

a parallel axis located halfway between them. Calculate the components of the 
quadrupole moment and find the total radiation. (Van Vleck.) 

18. A positive and negative charge are bound together by quasielastic forces to 
form a harmonic oscillator. Show that the radiation losses can be accounted for by a 
frictional force which, however, is proportional to the rate of change of acceleration 
rather than to the rate of change of displacement. 

19. A positive point charge oscillates with very small amplitude about a fixed neg- 
ative charge of equal magnitude. Show that the formulas for the field of an oscillating 
dipole can be obtained directly from Eqs. (35) and (37) on page 475 which apply to 
an accelerated point charge. 



CHAPTER IX 
BOUNDARY-VALUE PROBLEMS 

It can now be assumed that the reader is familiar with the principles 
that govern the generation of electromagnetic waves and the manner in 
which they are propagated in a limitless region of homogeneous, isotropic 
space. The " boundless region" is, of course, simply an abstraction. 
In point of fact the most interesting electromagnetic phenomena are 
those induced by surfaces of discontinuity or rapid change in the physical 
properties of the medium. These boundary phenomena are roughly 
of three types. Suppose that a wave, propagated in one medium, is 
incident upon a surface of discontinuity marking the boundary of 
another. In the first case the linear dimensions of the surface measured 
in wave lengths are very large. A fraction of the incident energy is 
reflected at the surface and the remainder transmitted into the second 
medium. The direction of propagation is in general modified and this 
bending of the transmitted rays is referred to as refraction. The laws 
that govern the reflection and refraction of electromagnetic waves at 
surfaces of infinite extent are relatively simple. If, however, any or all 
the dimensions of the surface of discontinuity are of the order of the 
wave length, the difficulties of a mathematical discussion are vastly 
increased. The perturbation of the primary field under these circum- 
stances is referred to as diffraction. In both cases the secondary field of 
induced charges and polarization is excited by a primary wave of inde- 
pendent origin. Both are inhomogeneous boundary-value problems as 
defined first for the static field on page 195. 

The third case referred to above is the homogeneous problem. A 
conducting body is embedded in a dielectric medium. Charge is dis- 
placed from the equilibrium distribution on the surface and then released. 
The resulting oscillations of charge are accompanied by oscillations in 
the surrounding field. This field can in every case be represented as a 
superposition of characteristic wave functions whose form is determined by 
the configuration of the body and whose relative amplitudes are fixed by 
the initial conditions. Associated with each characteristic function is a 
characteristic number that determines the frequency of that particular 
oscillation. The oscillations are damped, partly due to the finite con- 
ductivity of the body and partly as a result of energy dissipated in radia- 
tion. However, the positions of conductor and dielectric relative to the 
surface of separation can be interchanged. Electromagnetic oscillations 

482 



Sue. 9.1] BOUNDARY CONDITIONS 483 

then take place within a dielectric cavity bounded by conducting walls. 
If the conductivity is infinite, there is neither heat loss nor radiation and 
the oscillations are undamped at all frequencies. 

GENERAL THEOREMS 

9.1. Boundary Conditions. A conventional proof of the conditions 
satisfied at a boundary by normal and tangential components of the 
field vectors was presented in Sec. 1.13. In Chaps. Ill and IV the rela- 
tion of these boundary conditions to the discontinuous properties of 
certain integrals was established for the stationary regime; on the basis 
of Sees. 8.13 and 8.14, the same procedure can be extended to the variable 
field. We shall forego this analysis but shall give some further attention 
to the important case in which the conductivity of one of the two media 
approaches infinity. 

At the interface of two media the transition of the tangential com- 
ponent of E and the normal component of D is expressed by 

(1) n X (E 2 - Ei) =0, n (D a - DO = 6, 

where by previous convention n is the unit normal directed from the 
medium (1) into (2) and d denotes the surface charge to avoid confusion 
with a? = 2irv. The flow of charge across or to the boundary must also 
satisfy the equation of continuity in case cither or both the conductivities 
are finite and not zero. 

(2) n.(J.-J.) = ~ 

Suppose now that the time enters only as a common factor exp( zcoZ) 
and that apart from the boundary the two media are homogeneous and 
isotropic. Then (1) and (2) together give 

/0\ 2^2n *lEin = 5, 

(TzEzn VlEln = Uo5. 

Let us see first under what circumstances the surface charge can be zero. 
If d = 0, the determinant of (3) must vanish and hence 

(4) CTi 2 CT 2 i = 0. 

If 5 is not zero (and this is the usual case), it may be eliminated from (3) 
and we obtain as an alternative boundary condition on the normal component 
ofE: 

(5) Ml&!#2n - Ml^ln = 0. 

If either conductivity is infinite, (5) becomes indeterminate; but from (3) 

(Tj 

d, 



484 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

and in the limit as <r\ > <*> , 

(7) E ln - 0, # 2n -> - 5, 

2 

From the field equations, moreover, 

(8) Ei = - . V X Hi, Hi = -- V X 

v ' en 



so that, if by hypothesis the field intensities are bounded, both EI and 
Hi approach zero at all interior points of (1) as o-i <*>. The transition 
of the tangential component of E is continuous across the boundary and, 
consequently, in the case of an infinitely conducting medium 

(9) n X E 2 = n X E! = 0. 

Consider now the behavior of the magnetic field at the boundary. 
In general 

(10) n (B 2 - Bi) =0, n X (H 2 - Hi) = K, 

where the surface current density K is zero unless the conductivity of 
medium at the boundary is infinite. It has just been shown that Hi 
vanishes if <TI becomes infinite and in this case 

(11) n B 2 = 0, n X H 2 = K. 

A further useful boundary condition on the magnetic field in the case 
of perfect conductivity can be derived as follows. 1 Let the boundary 
surface S defined by the equation, 

(12) ? = fi(x, y, z) = constant, 

coincide with a coordinate surface in an orthogonal system of curvilinear 
coordinates , r;, f as in Sec. 1.16. If <n > oo , the tangential component 
of E 2 and the tangential component of V X H 2 , which is proportional to 
it, approach zero. Therefore by (80), page 49, 

(13) 

+ *i [it 



The normal coordinate is f. Since a\ is infinite, H[ is zero; hence, since 
the coefficients of the unit vectors must vanish independently, we have 
just outside the conductor 



(14) (/*,#,) =0, (h&d = 0. 



The quantities h^H^ and h^H^ are covariant components (page 48) of 
the vector H 2 tangent to the surface. The boundary condition is, there- 
fore: the normal derivatives of the covariant components of magnetic field 
1 The author is indebted to E. H. Smith for the proof. 



SEC. 9.1] BOUNDARY CONDITIONS 485 

tangent to the boundary vanish as the conductivity on one side becomes 
infinite. 

Under the head of "boundary conditions" we shall consider also 
the behavior of the field at infinity. In this respect the variable field 
differs notably from the static. If in the stationary regime all sources 
are located within a finite distance of the origin, the field intensities 
vanish at infinity such that lim R 2 E and lim R 2 H are bounded as R > oo , 
where R is the radial distance from the origin. The scalar potential 
satisfies V 2 < = and lim R<j) is bounded as R > oo . Every function < 
which is regular at infinity and which satisfies Laplace's equation at all 
points of space (no sources) is necessarily zero. On the other hand, if 
variable sources are located within a finite distance of the origin, the field 
intensities vanish such that lim RE and lim RK are bounded as R > oo . 
The scalar potential and rectangular components of the vector potential 
and the field intensities all satisfy the equation, 

(15) ' VV + &V = 0, 

at points where the source density is zero. Moreover there exist func- 
tions \f/ satisfying (15) throughout all space and vanishing at infinity 
which are not everywhere zero. 

A solution of Laplace's equation is uniquely determined by the sources 
of the potential and the condition that it shall be regular at infinity. 
These two conditions are not sufficient to determine uniquely the wave 
function ^, for (15) admits the possibility of convergent as well as divergent 
waves. This question has been investigated by Sommerfeld 1 in con- 
nection with the Green's function of (15) for spaces of infinite extent. 
To conditions that are analogous to those of the static problem must be 
added a "radiation condition." The problem is formulated as follows: 
The density g(x, y, z) of the source distribution is specified, and these 
sources are assumed to lie entirely within a domain of finite extent. 
Then ^ is uniquely determined if: 

(a) at all points exterior to a closed surface S (which can, if necessary, 
be resolved into a number of separate closed surfaces), ^ satisfies 

(16) W + *V = -0; 

(6) \l/ satisfies homogeneous boundary conditions over S of the type 

d\l/ 
at + ^ = 0, where a and are constants or specified functions of 

position; 

(c) ^ vanishes in such a way that lim R$ is bounded as R oo , a con- 
dition we shall again refer to as regularity at infinity; 

1 SOMMERFELD, Jahresber. deid. math. Ver. t 21, 326, 1912. 



486 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

(d) \l/ satisfies the radiation condition 

< 17) ?. R (H - 

which ensures that at great distances from the source the field represents 
a divergent traveling wave. It has been tacitly assumed that the time 
enters explicitly in the factor exp( iwi). 

The significance of (17) may be made clear by applying Green's 
theorem to (16) within a region bounded internally by S and externally 
by a surface SQ. Then, as in Sees. 8.1 and 8.13, 



(18) y . 

** *" , 



The first two terms to the right of (18) represent traveling waves diverging 
from, the source. The third term, however, expresses the sum of all waves 
traveling inwards from the elements of >S and must, therefore, vanish as 
iS u recedes to infinity. If this third term be designated by U, we have 

- ** ^) Tf da 

where d 12 is an element of solid angle. The second integral in (19) is 
extended over the finite domain 4?r and vanishes as R > oo provided ^ 
is regular at infinity as prescribed in (c) above. In order that U shall 
vanish, it is sufficient that 

(20) lim 

v ' fl->oo 

If at great distances SQ is replaced by a sphere of radius R, (20) is identical 
with (17). 1 

9.2. Uniqueness of Solution. Let V be a region of space bounded 
internally by the surface S and externally by $ . The surface S can be 
resolved, if the case demands, into a number of distinct closed surfaces 
Si as in Fig. 18, page 108. Then V is multiply connected (page 226) 
and the surfaces Si represent the boundaries of various foreign objects in 
the field. It will be assumed for the moment that the properties of V 
are isotropic but the parameters ju> *, and a can be arbitrary functions 
of position. Now let EI, Hi, and E 2 , H2 be two solutions of the field 
equations which at the instant t = are identical at all points of V. 
We wish to find the minimum number of conditions to be imposed on the 

1 An alternative proof was given in Sec. 8.13, p. 461. 



SEC. 9.2] UNIQUENESS OF SOLUTION 487 

components of the field vectors at the boundaries S and S in order that 
the two solutions shall remain identical at all times t > 0. 

In virtue of the linearity of the field equations (we exclude ferro- 
magnetic materials) the difference field E = E 2 EI and H = H 2 HX 
is also a solution. One may assume without loss of generality that the 
sources of the fields lie entirely outside the region V, since it was shown 
in the preceding chapter that the field is uniquely determined when the 
charges and currents are prescribed. Or, if sources appear within 7, 
one must specify that the distribution and rate of working of the electro- 
motive forces are in both cases the same. Then within 7, by Poynting's 
theorem, page 132, the difference field satisfies 



In order that the right-hand member of (21) shall vanish, it is only 
necessary th&t ^either the tangential components of E t and E 2 , or the 
tangential components of HI and H 2 be identical for all values of t > 0; 
for then either nxE = OornxH = and E X H has no normal 
component over the boundaries. In that case we have 

(22) 

The right-hand member of (22) is always equal to or less than zero. 
The energy integral on the left is essentially positive or zero and vanishes 
at t = 0. Hence (22) can only be satisfied by E = E 2 EI = 0, 
H = H 2 HI == for all values of t > 0, as was to be shown. An 
electromagnetic field is uniquely determined within a bounded region V 
at all times t > by the initial values of electric and magnetic vectors through- 
out 7, and the values of the tangential component of the electric vector (or 
of the magnetic vector) over the boundaries for t > 0. 

If >S recedes to infinity, 7 is externally unbounded. To ensure the 
vanishing of the integral of a Poynting vector over an infinitely remote 
surface, it is only necessary to assume that the medium has a conductivity, 
however slight. If the field was initially established in the finite past, 
the difficulty may also be circumvented by the assumption that S Q lies 
beyond the zone reached at time t by a field propagated with a finite 
velocity c. The theorem just proved does not take full account of this 
finiteness of propagation. We have established that the values of 
E and H are uniquely determined throughout 7 at time t by a tangential 
boundary condition and the initial values everywhere in 7. Physically 
it is obvious, however, that this is more information than should be 
necessary. The field is propagated with a finite velocity and, conse- 
quently, only those elements of 7 whose distance from the point of 



488 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

observation is <> (t )c need be taken into account. The classi- 
cal uniqueness proof given above has been extended in this sense by 
Rubinowicz. 1 

The theorem applies also to anisotropic bodies. Electric and mag- 
netic energies are then positive definite forms (cf. page 141), which is to 
say that they are positive or zero for all values of the variables. The 
proof remains unmodified. 

There is one aspect of the theorem which at first thought may be 
puzzling. We have seen that the field is determined by either the values 
of n X E or n X H on the boundary, yet in the problems to be discussed 
shortly we find it necessary to apply both boundary conditions, 

(23) n X (E, - Ei) =0, n X (H 2 - Hi) = 0, 

where here E 2 and EI denote values of field intensity on either side of the 
boundary. The reason for the apparent discrepancy is, of course, that 
n X E and n X H refer to the tangential components of the resultant 
field on one side of the surface. These are in general unknown and it is 
the object of a boundary-value problem to find them. Equations (23) 
simply specify the transition of the field across a surface of discontinuity 
and the two together enable us to continue analytically a given primary 
field from one region into another. Having thus determined the total 
field, the uniqueness theorem shows that there is no other possible 
solution. If, however, one side of a boundary is infinitely conducting 
we do know the tangential component of the total field, for in this case 
n X E = 0, and a single-boundary condition is sufficient for the solution 
of the problem. 

9.3. Electrodynamic Similitude. Since Newton's time the principle 
of similitude and the theory of models have had a most important influ- 
ence on the development of applied mechanics. This is particularly true 
of ship and airplane design which is governed very largely by the data 
obtained from small models in towing tanks and wind tunnels. It is 
customary to express the conditions of an experiment in terms of certain 
dimensionless quantities such as the Reynolds number. Thus the 
results of a single measurement of a given model can be applied to a series 
of objects of identical form and differing only in scale, provided the 
viscosity of the fluid and the velocity of flow are varied in such a way 
as to keep the Reynolds number constant. 

Similar principles prove very helpful in the design of electromagnetic 
apparatus. We shall write first the field equations in a dimensionless 
form. In a homogeneous, isotropic conductor 

\TT aTf 

(24) VXE + M ^ = 0, VXH-^~E = 0. 

dt ot 

1 RUBINOWICZ, Physik. Z., 27, 707, 1926. 



SBC. 9.31 ELECTRODYNAMIC SIMILITUDE 489 

Now let 

E eE, H = AH, 

length = ZoL, time = UT, 

where E, H, K, /c m , 5, I/, and T are the dimensionless measure numbers of 
the field variables in a system for which the unit quantities are e, A, 
co, MO, Zo, O-Q, and Jo. The measure numbers satisfy the equations 

V X E+a Km ~ = 0, 
(26) d ^ 

V X H $K, ^ ysE = 0, 

where 

are dimensionle&s constants. Upon eliminating the common ratio e/A, 
one obtains 

/l\* p 

(28) ji o (7-) = constant, MO<TO ~ = constant. 

Vo/ to 

From the first of these it follows at once that the product /z must have 
the dimensions of an inverse velocity squared. This is, no doubt, the 
most fundamental approach to the problem of units and dimensions. 1 

In order that two electromagnetic boundary-value problems be 
similar, it is necessary and sufficient that the coefficients OJC TO , j9je, and ys 
be identical in both. For t Q let us take for example the period r of the 
field, and for Z any length that characterizes one of a family of bodies 
which differ only in scale. Thus Z may be the radius of a set of concen- 
tric spheres, or the major axis of a set of ellipsoids. The condition of 
similitude requires that the two characteristic parameters Ci and C 2 in 

(29) A 

be invariant to a change of scale. Suppose that the characteristic length 
Zo is halved. Then both Ci and C 2 remain unchanged if the permeability 
n at every point of the field is quadrupled. This is an awkward remedy 
from a practical standpoint, but it is the only way in which the initial 
state can be simulated through the adjustment of a single parameter. If 
n and are left as they were, constancy of Ci results also from halving 
the period, or doubling the frequency, but this alone does not take care 
of C 2 . In order that the half-scale model shall exactly reproduce the 

1 Cf. Sees. 1.8. 4.8, and 4.9. 



490 



BOUNDARY-VALUE PROBLEMS 



[Ciur. IX 



full-scale conditions, it is necessary also that the conductivity be doubled 
at every point. 

This principle can be illustrated by reference to the type of high- 
frequency radio generator that maintains standing electromagnetic waves 
in a cavity resonator bounded by metal walls. The frequency of oscilla- 
tion is determined essentially by the dimensions of the cavity, while 
the losses depend largely on the conductivity of the walls. If the dimen- 
sions are halved, the frequency will approximately be doubled; if the 
conductivity of the walls is unchanged, it is entirely possible that the 
resulting increase of the relative loss will pass the critical value, so that 
the half-scale apparatus fails to oscillate. 

REFLECTION AND REFRACTION AT A PLANE SURFACE 
9.4. Snell's Laws, Two homogeneous, isotropic media have as a 
common boundary the plane S, and are otherwise infinite in extent. The 
unit vector n is normal to the plane S and directed from the region 

7 s 



Medium (2) 



Medium (1) 




FIG. 89. Reflection and refraction at a plane surface S. 

(ii Mi, on) into the region (c 2 , ^2, 0*2). Let be a fixed origin, which for 
convenience we locate on S. Then, if r is the position vector drawn from 
to any point in either (1) or (2), the interface S is defined by the equation 
(1) n-r-0. 

A plane wave, traveling in medium (2) is incident upon S. By (27), 
page 272, 



(2) 



n X E t , 



where E is the complex amplitude of the incident wave and n a unit 
vector which fixes its direction of propagation, 1 as in Fig. 89. The plane 
defined by the pair of vectors n and n is called the plane of incidence. 
1 It is apparent that so far as the present problem is concerned it would be neater 



SEC. 9.4] SN ELL'S LAWS 491 

The continuation of the primary field into medium (1) is determined 
by the boundary conditions at S. To satisfy these boundary conditions, 
a reflected or secondary field must be postulated in (2). Physically it is 
clear that the primary field induces an oscillatory motion of free and 
bound charge in the neighborhood of S, which in turn radiates a secondary 
field back into (2) as well as forward into (1). We shall make the 
tentative assumption that both transmitted and reflected waves are 
plane, and write 

E< - Ei *mi-'-*, H< = A ni x E,, 
(3) 7 

E r = E 2 ea-rr-o-S H r = ^ n 2 X E r , 

COM2 

where the unit vectors ni and n 2 are in the directions of propagation of 
transmitted and reflected waves respectively, and EI and E 2 are complex 
amplitudes, all as yet undetermined. By hypothesis, EI, E 2 , and E 
are independent of the coordinates and, consequently, if the tangential 
components of the resultant field vectors are to be continuous across S, 
it is necessary that the arguments of the exponential factors in (2) and 

(3) be identical over the surface n r = 0. But 

(4) r = (n r)n - n x (n x r) ; 

hence, at any point on the interface r = n X (n X r). Therefore, 
fc 2 n n X (n X r) = fc 2 n 2 n X (n X r), 



, . 

^ J fc 2 n n X (n X r) = /fcini n X (n X r), 



or, since n n X (n X r) = (n X n) (n X r), 

, fi v (n X n n 2 X n) n x r = 0, 

W (fc 2 n X n - fcini X n) n X r = 0. 

From these two relations it follows that n, n , ni, and n 2 are all coplanar. 
The planes of constant phase of both transmitted and reflected waves are 
normal to the plane of incidence. From the first of (6) also 

(7) sin 62 = sin (IT ) = sin , 

whence the angle of incidence is equal to the angle of reflection 2 . 
From the second of (6) 

(8) k z sin = ki sin 0i. 

Equations (7) and (8) express Snell's laws of reflection and refraction. 

from a notational point of view to let the incident wave travel from (1) into (2). 
Shortly, however, the plane S will be replaced by a closed surface S bounding a com- 
plete body. By previous convention n is directed outward from a closed surface and 
from medium (1) into (2). The choice of n as above will facilitate the comparison 
of formulas from the present with those of later sections. 



492 BOUNDARY-VALUE PROBLEMS [CiiAp. IX 

9.6 FresnePs Equations. The boundary conditions will now be 
applied to determine the relation between the amplitudes E^ EI and E 2 . 
At all points on S 

(9) n X (E + E 2 ) = n X EI, n X (H + H 2 ) = n X HL 

By virtue of (2) and (3) the second of these two can be expressed in terms 
of the electric vectors. 

(10) n X (n X Eo + n 2 X E 2 ) ~ 3 = n X (n x X EI) ^ 

M2 Ml 

Expansion of (10) leads to such terms as 

(11) n X (n X Eo) = (n E )n - (n n )E . 

The orientation of the primary vector E is quite arbitrary but can always 
be resolved into a component normal to the plane of incidence and conse- 
quently tangent to /S, and a second component lying in the plane of 
incidence. (Cf. Sec. 5.4, page 279.) The analysis is greatly simplified 
by a separate treatment of these two components of the incident wave. 
Case I. Eo Normal to the Plane of Incidence. Then 

n. EO == HO EO = 0. 

Since the media are isotropic, the induced electric vectors of the trans- 
mitted and reflected waves must be parallel to EO and hence also normal 
to the plane of incidence, so that n EI = n E 2 = 0. From Fig. 89 

n n = cos (IT ) = ~ cos , 

(12) n ni = cos (?r X ) = cos 0i, 
n H2 = cos 2 . 

Upon multiplying the first of Eqs. (9) vectorially by n and making use of 
(11) and (12), we find that the amplitudes must satisfy 

EO + E 2 = EI, 
^ ' cos E - cos 2 E 2 = ~ cos 



The relative directions of electric and magnetic vectors for this case are 
shown in Fig. 90. Solving (13) for EI and E 2 in terms of the primary 
amplitude E leads to 

E = M1&2 (cos 02 + cos ) E 

(14) ' *, coe J, + *i cos fc - (whena . Eo = ), 

= nJc* cos 0o pjci cos 0i E 

COS 02 + Ma&i COS 0i * 



These relations are not quite so simple as they appear at first sight, 
for 0i is complex if either (1) or (2) is conducting and may be complex 
even if both media are dielectric. By (7) and (8) 



FRESNEL'S EQUATIONS 



493 



SBC. 9.5] 

(15) cos 6 2 = cos , fci cos 0i = 

The angles of reflection and refraction can be eliminated from (14) and 
we obtain as an alternative form the relations 



COS 



(16) 



cos 0o + M2 V^i k\ sin 2 



.EO 



(when n EO = 0), 



__ /Ltifc 2 cos 0o M V&i - k\ sin 2 

. _: JB/0 

Mifc 2 cos + M2 V ^! k% sin 2 



Complex values of the coefficients of E imply that the amplitudes EI 
and E 2 themselves are complex and that the transmitted and reflected 
waves differ in phase from the incident wave. 




FIQ. 90. Polarization normal to the plane of incidence. 

Case II. EO in the Plane of Incidence. The magnetic vectors are 
then normal to the plane of incidence and parallel to S. 

n HO = n HI = n H2 

From (2) and (3) we have 



i X HI 
(17) J 2 " Kl 

2 = T 1*2 X H2, 

which when substituted into (9) give 

/- ox cos 0o Ho cos 2 H 2 == M cos 0i HI, 

(lO) HzKi 

HO + H2 = HI, 



494 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



as the conditions at the boundary. The relative directions of electric 
and magnetic vectors for this case are shown in Fig. 91. Solution of 
(18) leads to 

H = M2fci(cos 2 + cos ) H 

/I 2 fcj COS 2 + /iifc 2 COS 0i * 

(19) 

TT Mafcl COS 0o Mi&2 COS 0i . 

Xlo = " " 

jU 2 fci COS 0o + Ml&2 COS 0i 

or, upon elimination of 0i and 2 by (15), 

cos 0o 



(when n - H = 0), 



HI == 



cos 



H ( 



o> 



(20) 



k$ sin 2 

(when n HO = 0), 



i cos 



f fc^ sin 2 



nJc{ cos 0o + nikz *v fc? ~~ k% sin 2 0o 

When the incidence is normal, = 0, the two cases cannot be dis- 
tinguished and the amplitudes of transmitted and reflected waves reduce 
to 



(21) 



17 

Xl/2 = 



The relations expressed by Eqs. (14) and (19) were first derived in a 
slightly less general form by Fresnel in 1823 from the dynamical prop- 
erties of a hypothetical elastic ether. 

7 s 




FIG. 91. Polarization parallel to the plane of incidence. 

9.6. Dielectric Media. We shall study first the case in which the 
two conductivities <r\ and cr% are zero, so that both media are perfectly 
transparent. The permeabilities will differ by a negligible amount from 
Mo and SnelPs law can now be written 



SBC. 9.6] DIELECTRIC MEDIA 495 

sin 0i sin 



__ 
2 ~ \ = " ni2 ' 

where t>j and t> 2 are the phase velocities and n i2 is the relative index of 
refraction of the two media. If ei > c 2 it follows that n n < 1 and there 
corresponds to every angle of incidence a real angle of refraction 
0i. If, however, ei < c 2> as is the case when a wave emerges from 
a liquid or solid dielectric into air, then 0i is real only in the range for 
which nw sin ^ 1. The phenomena of total reflection occur when 
ni 2 sin > 1. We shall exclude this possibility for the moment and 
consider the Fresnel laws for real angles between the limits and Tr/2. 
When E is normal to the plane of incidence we now obtain from (14) 

_ 2 cos sin 0i _ 
* l ~ sin (0! + 0o) E ' 

(23) (n E = 0), 

T? - sin (0! - ) 

** " sin (0i + ) Eo> 

and for the components of E lying in the plane of incidence from (19) 

v 1? - 2 cos 0o sin X v> T? 

ni X El - sin (0 + 00 cos (0 - 0i) n X Eo ' 

(24) (n-H = 0), 
n v p _ tan (0o - 0Q 

n * x E * - teT07T~07 x ' 



Since the coefficients of E in (23) and (24) are real, the reflected and 
transmitted waves are either in phase with the incident wave, or out of 
phase by 180 deg. It is apparent that the phase of the transmitted wave 
is in both cases identical with that of the incident wave. The phase of the 
reflected wave, however, will depend on the relative magnitudes of 
and 0i. Thus if ci > e 2 , then 0i < , so that E 2 is opposed in direction 
to Eo in (23) and therefore differs from it in phase by 180 deg. Under 
the same circumstances tan (0 0i) is positive, but the denominator 
tan (0 + 0i) becomes negative if 0o + 0i > Tr/2 and the phase shifts 
accordingly. 

The mean energy flow is given by the real part of the complex Poynting 
vector. In optics this quantity is usually referred to as the " intensity " 
of light, but the term is ambiguous since it is also applied to the amplitudes 
of the fields. In the present case 



X (n X Eo) 
(25) 

V l 



l T72 Q 2 rr 2 

* -- ~ l ' ~ o * 



496 BOUNDARY-VALUE PROBLEMS [CH<U. IX 

Now the primary energy which is incident per second on a unit area of 
the dielectric interface is not * but the normal component of this flow 



vector, or n * = -^^ El cos . Likewise the energies leaving a 

Zi 

unit area of the boundary by reflection and transmission are 



(26) n S r = ^r E\ cos , n S, = -~ El cos 0i. 

t & 

According to the energy principle the normal component of energy flow 
across the interface must be continuous, 

(27) n (S + S r ) = n S, f 
or 

(28) -v/2 El cos 60 = \/7i El cos 0i + \^E\ cos . 

It can be easily verified that the Fresnel formulas (23) and (24) satisfy 
this condition. The reflection and transmission coefficients are defined by 
the ratios 

p - n*S r __ E\ n.< _ cos O l 

' 



B + r = i. 

In case E is normal to the plane of incidence, these coefficients are 

/OA N r> sin 2 (0i - ) sin 20 sin 20i 

(30) Bj - = sin^ (0i + )' T = sin* (*i + 0o)' 

and in the case of E lying in the plane of incidence 

tan 2 (0o - 0Q sin 20 sin 20i 



{ . 
^ } 



tan 2 (0 + 0i) sin 2 (0 + 0i) cos 2 (0 - 



If the incidence is normal, == 0i = 0, and it follows from (21) and 
(22) that 



(32) ^/ 

m 4 \ 



tf 



There is only one condition under which a reflection coefficient is 
zero. As + 0i > ir/2, the tan (0 + 0i) > oo and in this case R\\ 0. 
The reflected and transmitted rays are then normal to one another 

(HI -n! = 0) and sin Si = sin f | - J = cos ; it follows from (22) 



SEC. 9.7] . TOTAL REFLECTION 497 

that 



(33) tan = J 2 - n M . 

\2 

The angle that satisfies (33) is known as the polarizing, or Brewster angle. 
A wave incident upon a plane surface can be resolved, as we have seen, 
into two components, one polarized in a direction normal and the other 
parallel to the plane of incidence. The reflection coefficients of the two 
types differ and, consequently, the polarization of the reflected wave 
depends upon the angle of incidence. In particular, if incidence occurs 
at the polarizing angle, the reflected wave is polarized entirely in the 
direction normal to the plane of incidence. Use is sometimes made of 
this principle in optics to polarize natural light, although in practice it is 
less efficient than methods based on double-refracting prisms. 

The optical indices relative to air are usually of the order of n 2 i = 1.5; 
at radio frequencies they may be very much larger with a corresponding 
increase in the polarizing angle. Thus in the case of water, n 2 i increases 
from about 1.33 to 9 at radio frequencies and the polarizing angle from 
53 to 84.6 deg., which is not far removed from grazing incidence. Irregu- 
larities of radio transmission over water can doubtless be attributed on 
certain occasions to this cause. One will note also that at the polarizing 
angle T\\ is unity and that this is the only condition under which all the 
energy of the primary wave can enter the second medium without loss by 
reflection at the surface. 

9.7. Total Reflection. We return now to the case excluded from 
Sec. 9.6 of transmission from medium (2) into a medium (1) whose index 
of refraction is less than that of (2). The formulas of Sec. 9.6 are valid 
whatever the relative values of 61 and 2 , but if 0o is such that 



ni2 sin = \ sin 0o > 1, 

\*i 

they can be satisfied only by complex values of 0i. Physically a complex 
angle of refraction implies a shift of phase and the appearance of an 
attenuation factor. 

Let us suppose, then, that sin 61 > 1. The cosine is a pure imaginary. 



(34) cos 0i = -== \/*2 sm2 0o 1 = -mi 2 \/sin 2 



The radical has two roots whose choice will be governed always by the 
condition that the field shall never become infinite. To simplify matters*. 
a bit, the reflecting surface S will be made to coincide with the plane 
x = 0, as in Fig. 92. All points of medium (1) correspond to negative 
values of x. The phase of the transmitted wave is, therefore, 



498 
(35) 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



( x cos 0i + z sin 0i) 
( tx \/sin 2 rcfi 



sin 0o), 



where it is assumed that ^i = M2 = MO, and the field intensity of the 
transmitted wave is 



(36) 
where 

(37) 



a = w 



E< *= Ei e ftix+it 
sin , co 



< o), 



2rr 



0o ~ 



The field defined by (36) vanishes exponentially as x > -^ oo , whicK 
confirms the choice of the positive root in (34). 



Medium (2) 



'////////S/////S////S/* 



Medium (1) 
Fio. 92. Total reflection from a surface coinciding with the yz-plane. 

The amplitudes of reflected and transmitted fields are next deter- 
mined from the Fresnel laws after elimination of 0i. From (14) we obtain 

, 2 cos 



(38) 



E2J. ~ 



cos 0o + i A/sin 2 nil 
cos t x/sin 2 - 



E 



'OJL> 



cos 0o + i \/sin 2 0o nfj 
and from (19) and (3), for polarization in the plane of incidence, 

_ v T? - 2n * 1 cos 

Hi x ii/ig = 

*)?. pn<a ft 

(39) 



cos 



. QO X EO 



V F - 
X Ursil 



COS ^ "" 



. / ; u '^ u r 

n! cos + t v sin 2 n| x 

Since the coefficients of E are complex, it is apparent that the trans- 
mitted and reflected waves are no longer in phase at the surface with 
the incident wave. The reflection coefficient according to (29) is 
R = E 2 E 2 /J, and it follows at once from (38) and (39) that 

(40) 72 j. = flu = 1, T = TI = 0. 

The intensity of energy flow in the reflected wave is exactly equal to the 



SEC. 9.7] TOTAL REFLECTION 499 

intensity in the incident wave; there is no average flow into the medium o/ 
lesser refractive index. The field intensity in medium (1), however, is by 
no means zero. There is, in fact, an instantaneous normal component of 
energy flow across the surface whose time average is zero ; the time aver- 
age of the flow within (1) parallel to the surface (i.e., in the direction of 
2) does not vanish. This latter component is unattenuated in the direc- 
tion of propagation but falls off very rapidly as the distance from S 
increases due to the factor exp fax. The surfaces of constant phase in 
the transmitted wave are the planes z = constant, which are normal to 
surfaces of constant amplitude, x = constant. It is clear that E*g and 
Hi|| have components along the z-axis, which is the direction of propaga- 
tion within (1). The excitation within medium (1) in the case of total 
reflection is a nontransverse wave (it may be either transverse electric or 
transverse magnetic, page 350) confined to the immediate neighborhood of the 
surface. 

This analysis gives no clue as to how the energy initially entered (l) f 
for it is based on assumptions of a steady state and of surfaces and wave 
fronts of infinite extent. Actually, the incident wave is bounded in both 
time and space. The total reflection of a beam of light of finite cross 
section has been treated by Picht 1 who showed that the average flow 
normal to the surface is in this case not strictly zero. Any change 
causing a fluctuation in the energy flow of the transmitted wave destroys 
the totality of reflection. 

Figures showing the course of the magnetic lines of force and the 
lines of energy flow in total reflection have been published by Eichenwald 
and by Schaefer and Gross. 2 

Finally, let us examine the relative phases of the reflected waves. In 
(38) and (39) write 



Then since 

a ib .. d b 

.7 = e , tan - == -> 
a + ib ' 2 a 

we have 

(43) tan ^ = - 



cos 2 nfi cos 5 

Suppose that the incident wave is linearly polarized in a direction that is 
neither parallel nor normal to the plane of incidence. We then resolve 
it into components and discover that the resultant reflected wave is 

1 PICHT, Ann. Physik, 3, 433, 1929. See also NOETHER, ibid., 11, 141, 1931. 
1 SCHAEFER and GROSS, Untersuchungen ttber die Totalreflexion, Ann Physik 
32, 648, 1910. 



500 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

formed by the superposition of two harmonic oscillations at right angles 
to one another and differing in phase by an amount 

(44) 6 = $u - JJL. 

The reflected wave is elliptically polarized, since by (43) the relative phase 
difference d is not, in general, zero. 



,,., . _ 

(45) tan - 



tan ~ tan -~ 



tan - tan = 



The phase shifts 5.L and 5|| both vanish at the polarizing angle 
0o = sin" 1 ri2i, and their difference 5 is also zero at grazing incidence, 
= ir/2. The relative phase 5 attains a maximum between these limits 
at an angle found by differentiating (45) with respect to and equating 
to zero. The maximum occurs when 

(46) sin* * - 



which, when substituted into (45), gives 
(47) tan 5=5 l " 



2n 2 i 

This property of the totally reflected wave was used by Fresnel to 
produce circularly polarized light. It is first necessary that the incident 
wave be linearly polarized in a direction making an angle of 45 deg. with 
the normal to the plane of incidence. The amplitudes E 2 and E 2 \\ 
are then equal in magnitude. The relative index n 2 i and the angle of 
incidence are next adjusted do that 5 = ir/2, or tan 6/2 = 1. Accord- 
ing to (47) this condition can be satisfied only if 1 n^ > 2n 2 i, or 
n 2 i < 0.414, or rii 2 > 2.41. In the visible spectrum such a minimum 
value of the index of refraction is larger than occurs in any common 
transparent substance. To overcome this difficulty, Fresnel caused the 
ray of light to be totally reflected twice between the inner surfaces of a 
glass parallelepiped of proper angle. In the radio spectrum, on the other 
hand, the index of refraction may assume very much larger values. Thus, 
in the case of a surface formed by water and air, ni 2 = 9, n 2 i = 0.11. 
The condition tan 5/2 = 1 is then satisfied by either of the angles 
0o = 6,5 deg. or = 44.6 deg. The latter figure has been confirmed 
by measurements at a wave length of 250 cm. 1 

9.8. Refraction in a Conducting Medium. The phenomena of reflec- 
tion and refraction are modified to a striking degree by the presence of a 

1 BEBGMANN, Die Erzeugung zirkular polarisierter elektrischer Wellen durch 
cinmalige Totalreflexion, Physik. Z.. 33, 582, 1932. 



SEC. 9.8] REFRACTION IN A CONDUCTING MEDIUM 501 

conductivity in either medium. The laws of Snell and Fresnel are still 
valid in a purely formal way but, as in the case of total reflection, the com- 
plex range of the angle 0i leads to a very different physical interpretation. 
Let us suppose that medium (2) is still a perfect dielectric but that 
the refracting medium (1) is now conducting. The propagation con- 
stants are defined by 



where ai and /3i are expressed in terms of ei, jui, and <n by (48) and (49), 
page 276. By Snell's law we have 

k a 

(49) sin 0i = 7- s i n ^o = 2 <! 02 ( a i "~ ^0 a ^ n ^>> 

which it is convenient to write as 

(50) sin 0i = (a ib) sin . 
The complex cosine is then 



(51) cos 0i = VI - (a 2 - 6 2 - 2afa) sin 2 = pe*. 

The magnitude p and phase 7 are found by squaring (51) and equating 
real and imaginary parts on either side. 

(52) p2 C 8 2y ** p ^ 2 C s2 ^ ~ !) ^ x -~ ( a2 ~" &2 ) sin2 *o, 
p 2 sin 2y = 2p 2 sin 7 cos 7 = 2o6 sin 2 . 

The phase of the refracted wave is, as in (35) and Fig. 92, 

(53) fcini r = (i + i$\)(~x cos di + z sin 0i) 

= xp(a\ cos 7 $1 sin 7) ixp(0\ cos 7 + i sin 7) 

+ 2(aai + &i) sin 0o + iz(a@i bai) sin . 

From (49) and (50) it is readily seen that (aot\ + bfti) sin = a 2 sin 
and (a0i 6ai) == 0. Within the conducting medium the transmitted 
wave is represented, therefore, by 

(54) Ei = EieP*+ t ' ( -*+ <M * 8in '- w<) , ( x < o)^ 

where 

. J> = p(]8i cos 7 + <*i sin 7) 

q == p(ai cos 7 /Ji sin 7). 

Note that the surfaces of constant amplitude are the planes px constant, 
the surfaces of constant phase are the planes qx + z sin OQ z = constant, 
and these two families do not f in general, coincide. Within (1) the field 
is represented by a system of inhomogeneous plane waves, as in the case of 



502 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



total reflection. The planes of constant amplitude are parallel to the 
reflecting surface /S; the direction of propagation is determined by the 
normal to the planes of constant phase. The angle \f/ made by this wave 
normal with the normal to the boundary plane (in the present instance 
the negative x-axis) is the true angle of refraction and is defined by 

f56) x cos ^ + z sin \l/ = constant, 



or 

(57) cos \l/ = 



sin \l/ = 



sn 



+ OL\ sin 2 r VgM- oil sin 2 

The relation of the planes of constant amplitude to the planes of constant 
phase is illustrated in Fig. 93. 



Medium (2) 



Medium (1) 



'//////////%/////////////// Z 




Plane of constant amplitude 

\ 

Wave\normal 



FIG. 93. Refraction at a plane, conducting surface. 

A modified Snell's law for real angles is expressed by (57). 



(58) 



The;quantity n(0 ) is a real index of refraction which now depends on the 
angle of incidence, a notable deviation from the law of refraction in 
nonabsorbing media. The phase velocity, defined as the velocity of 
propagation of the planes of constant phase, is 



(59) 



t>i(0o) ~ 



CO 



+ al sin 2 



Not only does this velocity depend on the angle of incidence, but as in the 
case of total reflection there are also components of field intensity in the 
direction of propagation. The field within the conductor is not strictly 
transverse. 



SBC. 9.8] REFRACTION IN A CONDUCTING MEDIUM 503 

The calculation of p, g, and n in terms of the constants of the media 
and the angle of incidence is a tedious but elementary task. From (52) 
and (55), together with the definition of a and b implicit in (49), one 
obtains the following relations: 

P 2 (0o) = i[-*J + R + ai sin 2 0o 



p\ + (a? - j8j - a| sin 2 ) 2 ], 
(60) g*(0 ) = aj - 0? - a\ sin 2 



/ a - - a sn 

a\ sin 2 



l + (a? - /? - 1 sin 2 ) 2 ]. 

From these a subsidiary set of relations, known as Ketteler's equations, 
can easily be derived. 



(61) 



COS 



The functional relation of n(0 ) to the angle of incidence and the 
constants i and pi expressed by (60) has been confirmed by measure- 
ments in the visible region of the light spectrum. 1 No direct connection 
exists, however, between the observed values of ai and pi at optical fre- 
quencies and the static or quasi-static values of the parameters ei, /zi, 
and cri. In fact ai and /3i can assume values at optical frequencies which 
at radio frequencies are possible only in densely ionized media. Thus Shea 
found for copper ai/aj = 0.48, a value less than unity, and i/a 2 = 2.61, 
instead of the exceedingly large value one might anticipate for such a 
good conductor. In this case the apparent phase velocity within the 
metal is greater than that of light. The anomalous behavior of these 
parameters at optical frequencies gives rise to some very interesting 
phenomena in the domain of metal optics, which lie beyond the limits we 
have imposed upon the present theory. 2 

Although there appear to be no experimental data available in 
support of Eqs. (60) at radio frequencies, there is every reason to believe 
them exact. We shall discuss only the case in which the conduction 
current in the medium is very much greater than the displacement 
current. Let 17 = cr/w. Then the assumption is that r?? >>> 1, and 
under these circumstances it will be recalled (page 277) that 

1 SHEA, Wied. Ann., 47, 271, 1892; WILSET, Phys. Rev., 8, 391, 1916. 

8 An excellent account of the optical problems of reflection and refraction is given 
by Konig in his chapter on the electromagnetic theory of light in the "Handbuch der 
Physik," Vol. XX, pp. 197-253, Springer, 1928. 



504 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

(62) 
From (60), 




but, since 17? is assumed to be very much larger than the maximum value 
of sin 2 , we obtain the approximate formulas 




hence, n > <*> as en < or w 0. At the same time 

(65) sin 

and ^ 0. As the conductivity increases or the frequency decreases, the 
planes of constant phase align themselves parallel to the planes of constant 
amplitude and the propagation is into the conductor in a direction normal to 
the surface. 

In the case of copper, cri = 5.82 X 10 7 mhos/meter, and it is obvious 
that ^ differs from zero by an imperceptible amount. Whatever the 
angle of incidence, the transmitted wave travels in the direction of the 
normal. The factor 

(66) 

measures the depth of penetration. It is characteristic of all skin-effect 
phenomena and gives the distance within the conductor of a point at 
which the amplitude of the electric vector is equal to 1/e = 0.3679 of its? 
value at the surface. In this distance the phase lags 180 dog. Since the 
value of d measures the effectiveness of a material for shielding purposes, 
it is interesting to know its order of magnitude. The table below gives 
the values of 5 in the case of copper for several frequencies. They are 
obtained from the approximate formula 6 = 6.6i>~* cm. 



cycles /sec. cm. 

60 0.85 

10' 0.21 

10 0,007 

The depth of penetration is decreased by an increase in permeability, 
but this is usually offset by the poor conductivity of many highly perme- 
able magnetic materials. 



SEC. 9.9] REFLECTION AT A CONDUCTING SURFACE 505 

The angle \f/ may be approximately zero for materials of much lower 
conductivity than the metals. In the case of sea water, 

<TI = 3 mhos/meter, *i = 81c , 
and we find 

(67) ijx ~ X 10 8 , n = ~ X 10 s , & = ^ me ters. 

v v *> v */ 

At v = 10 6 we see that n = 164, ^ < 0.35 deg., and 5 = 29 cm. The 
approximation is just valid at v = 10 8 , for then r? 2 = 45 1, and in this 
case n = 16.4, i < 3.5 deg., d = 2.9 cm. 

9.9. Reflection at a Conducting Surface. We shall examine next the 
phase and amplitude of the wave reflected at the plane interface of a 
dielectric and a conductor. By SnelPs law 



(68) ki cos 0i - V*i ~ fc| sin 2 , 

and from (60) and (61) it follows that 



pq = 

(69) g 2 p 2 = a? |8J i sin 2 , 

g 2 + p 2 = [4af/3 2 + (a 2 /3 2 a| s i n2 
It can easily be verified from these relations that 

(70) ki cos 
where 

(71) tan = ; 

2 g 

Upon substituting (70) into the Fresnel equation (14) for the reflected 
component of the electric vector polarized normal to the plane of inci- 
dence, one obtains 

b? 

/7<>\ -p /*n*2 w* ^0 A*2 V Q' 2 ~f~ P 2 ^ -r* 

V'^ ^21 = ^ il/0. 




COS 

The fraction must be rationalized to give amplitude and phase. Then 
(73) E 2J _ = p^e-^-L E O _L, 

and after a relatively simple calculation we find 

2 = Qu2g MI cos ) 2 
PI> 
ton A = 

- 1 



COS ) 2 
COS 



cos 2 0o - M Kg 2 + p 2 ) 

The other component of polarization is found in the same way but 
the computation is considerably more tedious. According to (20), 



606 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

the second Fresnel equation is 

** 

/TCN TT ^2 cos (a? |8i + 2ij8if) /ii2VV + P 2 e 2 

(75) Had. = t > -ttoj.. 

The lengthy process of rationalizing this expression may be skipped over 
and the result set down at once. We abbreviate 

(76) n 2 X E 2 |i = p|[e-^ll n X Eon, 
and find 



2 = * " Pi) COS 00 "" 



= 

P " [M2(ttf ~ 0!) cos + Mi<*2?] 2 + [2/i2ai0i cos 

+ P 2 - <xj sin 2 ) cos 

- 



tan d || = 

If the conductor is nonmagnetic, so that MI = M2, the expression for the 
amplitude pj| factors into the form deduced by Pfeiffer 1 in the course of 
his optical studies: 

(Q 2 cos ) 2 + p 2 (q 2 sin tan ) 2 + p 2 

p||2 - JL* t J- ! i -1 ; i- 

As in the case of total reflection, the two components of polarization 
are reflected from an absorbing surface according to different laws. 
Consequently an incident wave which is linearly polarized, but whose direc- 
tion of polarization is neither normal nor parallel to the plane of incidence, 
will be reflected with elliptic polarization. The polarization is determined 
by the ratio 
(79) le''<ir*jL> = pe**. 

^ ' PJL 

Only in the case of nonmagnetic materials do the expressions for p and 6 
reduce to a relatively simple form. If /xi = jx 2 , one finds after another 
laborious calculation that 

2 (<7 <*2 sin 0p tan ) 2 + p 2 
P "" (q + <*2 sin tan ) 2 + p 2 

' ' , 2 2 p sin tan 

tan o = 



a\ sin 2 tan 2 - (q 2 + p 2 ) 

The reflection coefficients are again defined as the ratio of the energy 
flows in the incident and reflected waves. Thus 

(81) RL = PJL 2 , fill = Pll 2 - 

In the case of normal incidence R = JB|| = -B, 



R, Diss. Giessen, 1912; Ktoia, Zoc. a'<., p. 242. Cf. also WILSBY, 
loc. cit., p. 393. 



SEC. 9.9] REFLECTION AT A CONDUCTING SURFACE 507 

The degree of polarization is commonly measured by the ratio 
(83) s = I^MJIHM 2 _ 1 - PV, 

v ; 2 ~ 



where g* = El\\/El, the ratio of the incident intensities. 

When one considers that the quantities q and p are functions of the 
angle of incidence as well as of all the parameters of the medium [Eqs. 
(60)], the complexity of what appeared at first to be the simplest of 
problems the reflection of a plane wave from a plane, absorbing surface 
is truly amazing. The formulas are far too involved to be understood 
from a casual inspection, and the nature of the reflection phenomena 
becomes apparent only when one treats certain limiting cases. Of 
these, one of the most important for electrical theory is that in which 

Case I. tjl = -~* 1. Then 
(84) 
as was shown on page 504, and 




cos 



2 
(85) (l + ^ co- ,)' + 1 



, * ^ A*l"2 2 COS 00 

tan 5 1 c^. -. 



**,-! 

Since jui/Vz == MI/MO = Kmi is the magnetic permeability of the conductor 
and 2 /<> = *ei is the specific inductive capacity of the dielectric, we may 
write 



(86) *==-' = = 2.11X10-, 



where v is the frequency. In the case of all metallic conductors 
C 1, and (85) reduces further to 

Pi. 2 ^ 1 2s cos , 
tan S.L ~ 2x cos . 

This approximation is obtained by applying the binomial theorem to 
(85), and it is valid as long as the square of Mi2/M2i can be neglected 
with respect to its first power. Likewise in the case of polarization 
parallel to the plane of incidence 



508 



BOUNDARY-VALUE PRdBLEMS 



[CHAP. IX 



(88) 



p|1 



2 _ 



2 cos 2 0o 2x cos 0o + x* 



2 cos 2 6<> + 2x cos + 

2x cos 

tan OH = -5 - r-r- 
11 x 2 cos 2 



Here the square term in x has been retained since at a certain value of 
the numerator of py 2 becomes very small. This critical angle is obvi- 
ously the analogue of the Brewster angle discussed above. The minimum 
value of p|| 2 corresponds to an angle of incidence satisfying the relation 
\/2 cos 0o = x. Consequently if the minimum is sharply defined the 
ratio MiAi can be determined directly from the observed reflecting 
properties of the material. 

It is clear from these results that the reflection coefficients of the 
metals are practically independent of the angle of incidence and differ 
by a negligible amount from unity even at the highest radio frequen- 
cies. Thus at an air-copper surface, K m i = 1, *2 = 1, <TI = 5.82 X 10 7 
mhos/meter, so that x = 2.77 X 10~ 8 ^/~v. In the case of iron the con- 
ductivity is about one-tenth as large and the permeability 1 may be of the 
order of 10 s , but x is still exceedingly small, even at wave lengths of a few 
centimeters. The formula 



(89) 



R = 1 - 4.22 X 10- 4 



to which (87) and (88) reduce at normal incidence, was verified in the 
infrared region of the spectrum by Hagen and Rubens. 2 

The results of their measurements are shown in the following table 
at X = 12/z and X = 25.5M, where I/A = 10~ 4 cm. The theoretical values 
with which they are compared were calculated from the electrostatic 
values of the conductivities. At wave lengths less than 12/x the devia- 





1 - R t % at X - 12 M 


1 - R, % at X = 25.5/i 


Metal 
















Observed 


Calculated 


Observed 


Calculated 


Ag 


1.15 


1.3 


1.13 


1.15 


Cu 


1.6 


1.4 


1.17 


1.27 


Al 








1.97 


1.60 


Au 


2.1 


1.6 


1.56 


1.39 


Pt 


3.5 


3.5 


2.82 


2.96 


Ni 


4.1 


3.6 


3.20 


3.16 


Sn 








3.27 


3.23 


Hg 








7.66 


7.55 



1 At ultrahigh frequencies the permeability may be very much smaller. 
pare the measurements by Glathart, Phys. Rev., 56, 833, 1939. 
a HAGJDN and RUBENS, Ann. Physik, 11, 873, 1903. 



Com- 



SBC. 9.9] 



REFLECTION AT A CONDUCTING SURFACE 



509 



tions become more marked and at optical wave lengths there is no 
correspondence. These figures are extremely significant. They imply 
that as long as we are concerned only with metals, the macroscopic electro- 
magnetic theory is valid for all wave lengths greater than about 10~ 3 cm., or 
frequencies less than about 3 X 10 18 cycles /sec. 

The reflection of radio waves from the surface of the sea or land gives 
results of another order. Consider first the case of an air-sea surface, 
for which /c m i = * e2 = 1, <TI = 3 mhos/meter, and x = 1.22 X 10~ 4 V7. 
Since the approximations (87) and (88) are valid only when x <C 1, we 
are limited now to frequencies less than 10 6 cycles/sec. Let us take, for 
example, a 600-kc. broadcast, fre- 
quency, X = 500 meters. Thenz = 
0.094. In Fig. 94 are plotted the 
reflection coefficients J?x and JR|| of 
waves polarized normal and parallel 
to the plane of incidence as func- 
tions of the angle of incidence. One 
will note that R j. > R\\, and that at 
86.2 deg. the coefficient jBj reaches 
its minimum. Hence, if the electric 
vector of the incident wave is linear- 
ly polarized in a direction that is 
neither normal nor parallel to the 
plane of incidence, the wave reflected 
from the surface of the sea will be 
elliptfcaHy polarized. Moreover, if 
the transmitting antenna is vertical, 




FIG. 94. Reflection coefficients of plane 
radio waves from the surface of the sea. 



the wave will be polarized in the plane of incidence; thus, if it meets 
the sea at a grazing angle, it may be almost completely absorbed. 

If now the frequency is increased, or if the wave is incident upon 
fresh water or earth for which the conductivity is usually of the order of 
10~ 4 or less, the approximations (87) and (88) no longer hold but the 
character of the reflection phenomena is not greatly modified. We have 
then to consider 

Case II. ti\ = jl 1. By (52) and (53), page 277, 
(90) ai^-VT lf fa~ 188.3 "" 



If the frequency is sufficiently high on ^> fa, Eqs. (60) above reduce to 
(91) P 2 ^ 0, ? 2 ~ a\ - al sin 2 . 



510 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



Then 



(92) 



y2 sin 2 



cos 



PII 



i *2 sin 2 
cos 0p 



cos 



*e2 sn 



cos 



*2 sn 



To this approximation, only the dielectric properties of the medium 
enter and (92) is identical with the results obtained in Sec. 9.6. The 
next order would introduce a small correction in fa. 1 



Let us take for example a fresh- water surface. /c 2 = 1, 
L ~ 2 X 10~ 4 mho/meter. Then 
4.45 X 10 4 



= 81, 



i c- 1.88 X 10-V, /3i ^ 4.18 X 10- 3 . 
v 

The approximation (92) holds, therefore, if v > 10 6 . The constants for 
earth and rock vary widely, but we may take as typical the case K e i = 6, 
0-1 ~ 10~ 5 . Then 



TfJl = 



3 X 10 4 



^ 5.1 X 10- 8 , fa z* 7.7 X 10~ 4 , 



so that again (92) is valid for v > 10 6 . At frequencies greater than 10 6 
cycles /sec., fresh water and dry earth or rock act as dielectrics in so far as 
their reflecting properties are concerned. The transmitted wave is of course 

rapidly attenuated due to the fac- 
tor fa. The reflection coefficients 
for these two cases have been plotted 
in Fig. 95 against angle of incidence. 
Such curves have been verified ex- 
perimentally by Pfannenberg. 2 

Although in the case of metals 
the formulas hold for frequencies 
extending into the infrared, this is 
not true for dielectric or poorly con- 
ducting materials. At wave 
lengths of the order of a few centi- 
meters most dielectrics exhibit a 
marked increase in absorption ow- 
ing to causes other than electrical 
conductivity. Thus at a wave 
length of 2.8 cm., the value of ft for 
distilled water computed on the basis 
of an electrical conductivity a = 2 X 10" 4 mho/meter is 4.2 X 10~ 3 , while 

1 An account of various approximation formulas useful in optics is given by Kdnig, 
loc. cit., p. 246. 

2 PFANNENBERG, Z. Physik, 37, 768, 1926. 




45 

Q , degrees 

FIG. 95. Reflection coefficients of plane 
radio waves from earth and fresh-water 
surfaces at frequencies greater than one 
megacycle per second. 



SEC. 9.10] REFLECTION AND TRANSMISSION COEFFICIENTS 



the measured value 1 is 494 (m.k.s. units, distance in meters!). This is an 
extreme case corresponding to dipole resonance of the water molecule, but 
it is obvious that under such circumstances the approximations (92) are 
invalid. The general formulas may always be applied, even in the optical 
spectrum, provided the values of a and ft are those determined by 
measurements at a specific frequency. 

PLANE SHEETS 

9.10. Reflection and Transmission Coefficients. We shall consider 
only the case of normal incidence, which is of considerable practical 

<*i> V/A^ 7 A (*,) 



E t ,H t 





FIQ. 96. Reflection and transmission of plane waves by a plane sheet at normal incidence, 
interest. Three arbitrary homogeneous media characterized by the 
propagation factors k\, k 2 , k z are separated by plane boundaries as shown 
in Fig. 96. We need deal only with the magnitudes of the vectors and 
write for the incident and reflected waves in medium (1) : 

(1) E- = E e ikiz ~ itat H- = ^V E. 



(2) 



E r = 



77 - 

n.r 



ti 



The field within the sheet (2) must be expressed in terms of both positive 
and negative waves. 



ff 2L (E+ fik 

* * TO V,-*-' Q ^ 

COJU 2 

while the transmitted wave is 

( A\ Z7* - -. T/i j-iikiz itdt 

(?) & t & 3 e K wt , 

It will be convenient to employ here the intrinsic impedance concept 
introduced in Sec. 5.6, page 282. For a plane wave in a homogeneous, 
isotropic medium 



H t = 



(5) 



Z = 



co/i,- 



1 BXz, Physik. Z., 40, 394, 1939. 



512 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

We shall now define the impedance ratios 

/ A \ 
(6) 

and note that 
(7) 

In terms of the impedance ratios the boundary conditions lead to 
four relations between the five amplitudes. 



Upon solving for the amplitudes of the reflected and transmitted waves, 
one finds 

w _ (1 ~ g)(l 

l 



We shall now define the complex ratios 



^ ~~ 1 j_ 7 ~~ *7 _|_ 17 ~~ ?' 

1 + ^,-fc Z^ + Zy 

The physical significance of these ratios is apparent when one notes that 



+ 0*rf/ + MA) 2 " ' 

The /2,-jfc are the reflection coefficients at normal incidence for the plane 
interface dividing two semi-infinite media [Eq. (82), page 506]; the 
quantity r jk is the complex ratio of the amplitudes of reflected and incident 
waves. The reflection coefficient at an interface is zero when the intrinsic 
impedances of the adjoining media are equal. 
In terms of these r#, (9) reduces to 



Eo (1 + Zu)(l + Z 23 ) 1 + r 12 r 23 
The reflection and transmission coefficients of the sheet are equal to the 



SEC. 9.11] APPLICATION TO DIELECTRIC MEDIA 513 

squares of the absolute values of these ratios. We write 

7 w I^^M 
"ik = */- 

(14) 

tan 7jfc = 



tan i = 
n * 



One will note that 7,* = 7*?, but 6,* = +$*/. Upon introducing these 
phase angles into (13), and then multiplying each ratio by its conjugate 
value, we obtain 




(16) 

K . 



E 



cos (* - 



(17) T = 



1 +2 

77T 2 

lo 



tn e~ 2f> ' d cos ( + 5 28 + 



+ 



M2/3 3 ) 2 ] 



1+2 V#i2# 2? e- 2 ^ d cos (612 
Equation (17) can be reduced to the form 
(18) T 

M3 a? + PI [(1 - Bn)(l ~ ^23) - 4 



by applying the formulas 



1 - fi/* 



(20) 



9.11. Application to Dielectric Media. The interference phenomena 
represented by (16) and (18) have frequently been applied to the measure- 
ment of the optical constants of materials in the infrared and electrical 
as well as the visible regions of the spectrum. 1 To make clear the 

1 See, for example, Drude, "Lehrbuch der Optik," Chap. II, 1900. The method 
has been used in the infrared by Czerny and his students. Czerny, Z. Physik, 66, 
600, 1930; Barnes and Czerny, Phys. Rev., 38, 338, 1931; Cartwright and Czerny, 
Z. Physik, 86, 209, 1933; Woltersdorff, ibid., 91, 230, 1934. Recently the same 



514 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



physical behavior of the reflection coefficient as the thickness of the sheet 
is varied, we shall treat the simple case of a thin dielectric sheet separating 
two semi-infinite dielectric media as in Fig. 96. The inductive capacities 
of the three media are 61, c 2 , 3. 0i ft 2 = = 0, and consequently 
a la = 5 23 = o. Then 



(21) 
(22) 



a k 




p = 



7*23) 2 




V e* 
sin 2 



r 

r /Jfc , 



sn 



and <* 2 = 27r/X 2 , where X 2 is Me wove fenj/tA within the sheet. Now if e 2 
lies between 1 and c 3 in value, the product ri 2 r 23 is positive and R oscillates 




FIG. 97. Reflection coefficients of plane dielectric sheets as functions of thickness and 

dielectric constant. 

as a function of d. The minimum value of R occurs at d = X 2 /4, and is 
zero when ri 2 = r 28 ; i.e., when 2 is the geometric mean of 1 and e 8 , or 
2 = V^a- Under these conditions all the energy is transmitted from 
medium (1) into medium (2), none is reflected at the surface. The intro- 
duction of a quarter-wave length sheet of the proper dielectric constant 
accomplishes the same purpose as the matching of impedance at the junction 
of two electrical transmission lines. In Fig. 97 curves are plotted which 
show the effect of a dielectric sheet at the interface of air and water. 

This principle has been applied recently to the manufacture of 
optically "invisible" glass. The reflection of light from a glass surface 
is reduced almost to zero by coating with a film of the proper index of 
refraction. 

method has been applied to ultrahigh-frequency radio waves. See for example 
Baz, Phys. Z. y 40, 394, 1939, and Pfister and Roth, Hochfreq. und Elektroak., 51, 166, 
1938, who have extended the formula for the reflection coefficient to arbitrary angles 
of incidence. 



SEC. 9.12] 



ABSORBING LAYERS 



515 



9.12. Absorbing Layers. If the medium on either side of the sheet 
is the same, Eqs. (16) and (18) reduce to 



[(1 



sin* 



- # 12 e- 2 ^) 2 + 4# 12 e~w* d sin 2 
A further simplification is achieved by defining 
(25) 5i 2 = -| In 72 12 , 

which when introduced into (23) and (24) leads to 

sin 2 aid + sinh 2 



(26) 
(27) 



n __ 
~ 




V 0,2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 

d in centimeters 

FIG. 98. Reflection coefficient of a plane sheet of water of thickness d cm., Xo = 9.35 cm., 
03 = 5.95 cm.- 1 , /3 2 = 0.456 cm." 1 . 

The reflection and transmission coefficients are again oscillating functions 
of the thickness d, but the amplitude of the oscillations is no longer con- 
stant. In Fig. 98, R is plotted against d for a thin sheet of water at a 
free-space wave length of 9.35 cm. The medium on either side of the 
sheet is air. We have previously noted that the attenuation factor of 
water in this region is very much larger than can be accounted for by 
the electrical conductivity alone. The observed values are a 2 = 595 
meters" 1 , 2 == 45.6 meters'" 1 . Within the sheet the wave length is 
X 2 = a X /a 2 = 1.05 cm., and the minima occur approximately at half- 
wave-length intervals. In general, the smaller the attenuation factor, 
the more nearly do the minima coincide with the half-wave-length points. 
In the case of commercial insulators 2 is at least 10 4 times as large as 2 , 
so that X 2 , and hence a^ can be determined directly from the observed 



516 BOUNDARY-VALUE PROBLEMS [Cnx*. IX 

oscillations of the reflection coefficient. The numerical values of the 
minima are determined by the losses, and a curve such as that of Fig. 98 
provides one of the best means of finding /3 2 at ultrahigh frequencies. 

In the case of a metal sheet the attenuation is so great, and the reflec- 
tion coefficient so nearly unity, that transmission, through even the 
thinnest sheets, is wholly negligible. The transmission coefficient 
through a copper sheet 1/z thick (10~ 4 cm.)? at 10 8 cycles/sec, for example, 
is less than 10~ 6 . However, this calculation gives a wholly erroneous 
impression of the effectiveness of shielding, for it applies only to plane 
surfaces of infinite extent. The reflection losses from surfaces whose 
radius of curvature is small compared to the wave length is by no means 
always large, 1 and the only significant general criterion is the value of 
the attenuation factor 2 , or its reciprocal, the skin depth, defined on 
page 504. 

SURFACE WAVES 

9.13. Complex Angles of Incidence. The possibilities of the Fresnel 
equations are far from exhausted by the discussion of the preceding 
sections, for it has been confined exclusively to real angles of incidence. 
These equations represent the formal solution of a boundary-value prob- 
lem which analytically is valid for complex as well as real angles. Com- 
plex angles of refraction already have occurred in connection with total 
reflection and reflection from conducting surfaces. There the associated 
plane waves proved to be inhomogeneous: the planes of constant ampli- 
tude fail to coincide with the planes of constant phase. Thus one may 
reasonably anticipate that a complex angle of incidence will be associated 
with an inhomogeneous primary wave. We shall not consider at this 
point how such waves are generated, but recall that in Sees. 6.8 and 7.7 
it was shown that cylindrical and spherical waves of the most general 
type can be represented as a superposition of plane waves whose direc- 
tional cosines include complex as well as real values. 

If the angle of incidence is real, the primary wave gives rise in general 
to a reflected wave at the surface of discontinuity. Only in the case of 
a plane wave polarized in the plane of incidence and meeting the inter- 
face of two semi-infinite dielectrics at the Brewster angle (page 497) does 
there occur an exception to this rule. There is always a reflected wave 
if one of the media is conducting. If, however, complex angles of inci- 
dence are admitted, the reflection coefficients can be made to vanish 
whatever the nature of the media. 

We shall consider in some detail the case of a wave whose electric 
vector is parallel to the plane of incidence. Letting r\\ represent as in 

1 A point well illustrated in the case of cylindrical shields by Schelkunoff, Bell 
System Tech. /., 13, 532, 1934. 



SEC. 9.13> 



COMPLEX ANGLES OF INCIDENCE 



517 



(15), page 513, the ratio of the reflected to the incident magnetic vector, 
the Fresnel equation (19), page 494, gives for this case 



(1) 
where 

(2) 



r 

11 



cos 0o - 



cos 



cos + Zu cos 



as defined on page 512. In order that there shall be no reflected wave 
the numerator of (1) must vanish. This condition, together with SnelTs 
law, leads to the two relations 



(3) 



cos 0i = 2 2 i cos , sin 0i = Zu sin , 



which determine the angles of incidence and refraction. Upon elimi- 
nating 0i one obtains 



(4) 



sin 2 6t> = 






12 



To simplify matters we shall assume that the interface of the two 
media coincides with the surface x = 0. The x-axis is normal to the 




Fia. 99. Refraction of the planes of constant phase characteristic of a surface wave. 

surface and is positive in the direction leading from medium (1) into 
medium (2), as shown in Fig. 99. The plane of incidence is parallel to 
the xz-plane; consequently, the field has only the one magnetic com- 
ponent H v . The electric vector lies in the plane of incidence and has in 




518 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

general both the components E 9 and E,. The field is transverse magnetic. 1 
Since the magnetic vector is tangent to the surface of discontinuity, its 
transition from one medium to the other is continuous; hence, by Sec. 9.4 

H y = Ce ik ~* coa e *+* 8in *)-<, ( x > 0), 

w H v = Ce'*^- xc08 ^+* flin ^)-^ ( x < o), 

where C is an arbitrary constant. 
Next we shall let 

(6) h = fc 2 sin do = &i sin 0i, 

and define the real and imaginary parts of the factors appearing in the 
exponents of (5) as follows: 

h = a + t'0, 

(7) i&2 cos 
t'fci cos 

The signs of the radicals have been so chosen that the field will vanish for 
infinite values of x. Then in terms of (7), 

,ox H v = C* 2 , (x > 0), 

W # = C^ t , (x < 0), 

where 

,g^ ^i = exp[+pix - Qz + i(-qix + z) torf], 

^2 = exp[ p 2 j^2 + i( q z x + az) t"co(|. 

Let us consider first the nature of the incident wave. We note that 
the planes of constant phase are defined by 

(10) q& + az = constant, 
while the planes of constant amplitude satisfy 

(11) p& ftz = constant. 

The planes of constant phase and constant amplitude are not coincident. 
The real angle of incidence \I/ Q is the angle made by the normal to the 
planes of constant phase with the positive z-axis. It satisfies the relation 

(12) x cos ^ + z sin \I/ Q = constant, 
whence 

(13) cos ^o = , *** > sin ^ = 



/ 
a 2 + ql V a 2 + gf 

The planes of constant phase are propagated in the direction of their 
positive normal with a phase velocity 

(14) t> 2 

1 See Sec. 6.1, pp. 



SEC. 9.13J 



COMPLEX ANGLES OF INCIDENCE 



519 



a velocity which is less than the normal velocity of a homogeneous plane 
wave in an infinite medium of the same character. On the other hand, 
the velocity parallel to the z-axis of a given point on a phase plane may be 
greater than the velocity in free space. The reason is clear from Fig. 100- 
The wave length X 2 is the distance between two planes of corresponding 
phase measured in the direction of the normal. 

(15) X 2 = 



The distances between the same two planes measured along the x- and 
z-axes, respectively, are 

(16) X, Xz 2 " ^ X * 2?r 



cos 



sin 



The phase point must travel these distances in the course of one period 
so that the apparent velocities are 



(17) 



CO 

-' 



CO 

-' 



and these quantities have values which lie between 

The normal to the planes of constant 
amplitude makes an angle \I/' Q with the 
z-axis, satisfying the relation 



and infinity. 



(18) x cos $Q + Z sir 
whence 



o = constant, 



(19) 



tan^ = - 




FIG. 100. The component of phase 
velocity parallel to the z-axis may be 
greater than the velocity of light. 



Planes of constant amplitude, which in 
the case of large conductivity are ap- 
proximately at right angles to planes of 
constant phase, have not been drawn in Fig. 99. 

Turning next to the transmitted wave in the region x < 0, we see 
from (9) that the planes of constant phase and constant amplitude are 
defined respectively by 

(20) qix + az = constant 

and 



(21) p 

whose normals make angles 



flz = constant, 

and \l/[ with the negative x-axis defined 



(22) 



j. i a 

tan wi = > 



tan rf = 1. 



520 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

The phase velocity is 

(23) vi 



The components of the electric vector are best calculated from the 
field equation 

(24) E = ^ V X H, 

which resolved into rectangular components gives 

_ tcoM dH v iw/i dH v 



From (8) and (9), it follows that 



( + 
(26) 2 . (* > 0), 



A/2 
COJUl / 



(27) l (* < 0), 



Physically, the field described by (8), (26), and (27) is associated with 
a current in the z-direction whose distribution with respect to y is uniform. 
Such a field is transverse magnetic, and the longitudinal component of 
electric field accounts for the losses in either medium, since E Z H V is the 
component of energy flow normal to the interface. The transverse com- 
ponent E x , on the other hand, gives rise to an energy flow parallel to the 
plane. The whole problem is comparable to the limiting case of a 
current propagated along a wire whose radius becomes infinite. The 
other symmetry, corresponding to the condition r = 0, leads to trans- 
verse electric waves and may be interpreted as the limiting problem of 
propagation along a solenoid of very large radius. In this case the cur- 
rent is transverse to the elements of the cylindrical surface. 

9.14. Skin Effect. If both media are perfect dielectrics, the angle 
is real in the case of TU = and is identical with the Brewster polarizing 
angle. Of much greater practical interest is the case in which medium 
(1) is a conductor and medium (2) a dielectric. One may then assume 
that |Z 12 | 2 1. According to (4) and (6) 



(28) A 2 = fcf 



,, 



SBC. 9.14] SKIN EFFECT 521 

which can now be expanded in powers of Zi 2 . 

(29) h = A 

of which we shall retain in the present approximation only the square 
term. Since by hypothesis the conductivity of (2) is zero, we may write 

2 2 1 

/q/Y\ 72 ^1 2//2&> /Xi 2 1 ~ irji 

i,OU; Z 12 ~2 A ' V . = -3 : r-> 

M 2 i/zior -f- KTiniw /i 2 ci 1 + Tjf 
where as above 771 = cn/eiw. Then 

(31) A - 

Note that a, the real part of this expression, is less than & 2 and that, 
consequently, the velocity in the z-direction of a point on a plane of 
constant phase is greater than that of a free wave in the dielectric. We 
Saw above that this is only an apparent velocity and that the true phase 
velocity is in fact less than that fixed by k 2 . 

We now make the further assumption that ijf 1, and that 1/r?? 
can, therefore, be neglected with respect to 1/rn. To this approximation 




(32) 

6V\ \ M2 

Thus the velocity in the z-direction is essentially the characteristic phase 
velocity of the medium and the attenuation approaches zero with 
increasing conductivity. 

To the same approximation 

(33) k\ 

which by (7) leads to 

(34) 

Likewise, 

(35) k\ - V 



(36) 

From these results one will note, first, that 

(37) 2L~?? 

9s P 



522 



BOUNDARY-VALUE PROBLEMS 



[CHAP. IX 



whence it appears from (13) and (19) that the planes of constant amplitude 
in the incident wave are normal to the planes of constant phase. As a\ > oo , 
tan ^o ~~* and ^o * fl"/2. As the conductivity increases, the planes of 
constant phase in the dielectric become very nearly normal to the surface of the 
conductor. 

Within the conductor, the corresponding relations are determined 
by (22). 



(38) 



tan \(/i ~ 



tan \l/[ ~ , 



Mi^i ., 

As <TI > oo 9 both ^i and \[/( approach zero. The planes of constant ampli- 
tude and constant phase within the conductor are very nearly parallel to the 
interface. At the same time the wave length becomes very small, for 



(39) 



Xi 



It will be recalled that in the case of metals <n is of the order of 10 7 
mhos/meter. The deviation of \I/ Q from ?r/2 and of \l/i from zero is then 
completely negligible, and the wave length in meters within the metal is 




Conductor 

FIG. 101. The solid lines represent planes of constant phase, the dotted lines planes of 

constant amplitude. 

in magnitude of the order v~*. Even in the case of sea water, at a 
frequency of 1 megacycle/sec., the forward tilt of the phase planes above 
the surface amounts to only some 10' of arc. The forward tilt of a radio 
wave propagated over dry earth may be more marked. The relation of 
the planes of constant phase and constant amplitude is then somewhat 
as illustrated in Fig. 101. It must be remembered, however, that in such 
a case the assumption r?f ^> 1 is not necessarily justified. 

Upon introducing the approximations (32), (34), and (36) into (26) 
and (27), we obtain for the field components the expressions 

(40) 
(41) 
Note that the tangential component E is continuous at the boundary 




SBC. 9.14] 



SKIN EFFECT 



523 



x = and equal in magnitude to (MIW/VI)* times the magnitude of the 
tangential component of magnetic field. As <n , the longitudinal 
electric field vanishes. 

The normal and tangential components of E differ in phase by 45 deg. 
(A more exact calculation shows this angle to be less than 45 deg. in 
the case of poor conductors.) Consequently the locus of the resultant 
electric vector at any point is an ellipse. At a point within the dielectric 
just above the surface, the ellipse is inscribed within a rectangle whose 
sides are 2a 2 = 2(jLt2/2)*, 26 2 = 20uia>/<ri)*, 62 <C a^ as shown in 

E x 



E, 




k- 2& 2 - 

FIG. 102a. 
Locus of the electric 
vector at a point in 
the dielectric. 



E, 



FIG. 102&. Locus of the 
electric vector at a point in the 
conductor. 



Fig. 102a. Since the particular value of z is of no importance, the 
coordinates of the ellipse are 



(42) 



E z = 02 cos ut, 



= 2 cos 



The electric vector in the dielectric rotates in a counterclockwise direction. 
The ellipse is identical in form at other points of the dielectric but 
reduced in size by reason of the attenuation factor. 

Within the conductor the electric vector describes an ellipse bounded 
by a rectangle whose sides are 2i = 2a>( 2 ju 2 )V cr i; 26 1 = 2(^ico/<ri)*, 
61 ^> ai. The coordinates of the ellipse are 



(43) 



E x = 



cos 



E 8 = 61 cos 



E z is now retarded with respect to E x and, consequently, within the 
conductor the electric vector rotates in a clockwise direction. 

The depth of penetration within the conductor is determined by the 
damping factor pi, whose reciprocal is exactly the factor 5 defined on 
page 504. The current density in the 2-direction is J n <riE g) and the 
total current passing through an infinite strip 1 meter in width parallel 
to the #?/-plane in the conductor is 



524 BOUNDARY-VALUE PROBLEMS (CHAP. IX 

(44) / = ^ J^ 1 J ^ E, dx dy = (?-"+ . 

On the other hand, the current through such a strip of depth d is 

(45) A/ = 7(1 - 6 -P*HW). 

Since p\ is very large in the case of good conductors, the current is con- 
fined almost entirely to a thin "skin" in the neighborhood of the surface. 
One will note, moreover, that the phase of a current filament varies 
with its depth. This effect may be expressed in terms of an internal 
self-inductance. 1 

PROPAGATION ALONG A CIRCULAR CYLINDER 

9.15. Natural Modes. A circular cylinder of radius a and infinite 
length is embedded in an infinite homogeneous medium. The propaga- 
tion constant of the cylinder is ki, that of the external medium k%. No 
restriction is imposed as yet upon the conductivity of either. The 
components of the field in circular coordinates were derived in Sec. 6.6. 
This field must be finite at the center and, consequently, the wave func- 
tions within the wire will be constructed from Bessel functions of the 
first kind. Outside the cylinder the Hankel functions H ensure the 
proper behavior at infinity. According to (36) and (37), page 361, we 
have at all interior points, r < a, 



< = 2 






</n(Xir) 6 - 



(i) Ef = - 2 f S J *M < + TT /:(M 6 " Fn> 

n mm 00 L l 

l/n(Xir) a ]Fn > 



= 2 



1 See the treatment by Sommerfeld in Riemann- Weber, "Differentialgleiehungen 
der Physik," Vol. II, pp. 507-511, 7th ed., 1927. 



SBC. 9.15] NATURAL MODES 525 

At all exterior points, r > a, 



' = 2 s^W):- 



(3) Ef - - 2 [$: fli'CX*-) < + ^ tf <"'(X,r) 6'] F n , 

o 

;= 2 tfTO*) atf 1 ., 



H ' = 

H '' = 2 



In these relations 

X? = kl - A 2 , X| = k\ - A 2 , 

(5) JF 7 * = exp(in0 + ihz iuf), 

and the prime above a cylinder function denotes differentiation with 
respect to the argument Xr. 

The coefficients of the expansions and the propagation factor A are 
as yet undetermined. However, at the boundary r a, the tangential 
components of the field are continuous, and this condition imposes a 
relation between the coefficients. From the continuity of the tangential 
components of E, we obtain 

(6) w 2 n ^ u ' a * ' u~ W n ~~ ~tf> n v*' 

J n (u) a' n = #i l) (tO <, 

while from the tangential components of H 

1 Tt / \ 1 fvfl T / \ T f v'v^ 

(7) ""~ - - -- ' - 



J n (u) 6 = Hi(i;) b' n , 
where 
(8) w = Xia, v = X 2 a. 

Equations (7) and (8) constitute a homogeneous system of linear 
relations satisfied by the four coefficients c, 6j>, a;, 6;. The system 



526 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

admits a nontrivial solution only in case its determinant is zero. The 
propagation factor h is determined by the condition that the determinant of 
(6) and (7) shall vanish. Expansion of this determinant leads to the 
transcendental equation 



fa J M 

U JnM 



- . 

i) W H?(V) 

whose roots are the allowed values of the propagation factor h and so 
determine the characteristic or natural modes of propagation. These 
roots are discrete and form a twofold infinity. The Bessel functions are 
transcendental; thus, for each value of n there is a denumerable infinity 
of roots, any one of which can be denoted by the subscript m. Any root 
of (9) can then be designated by h nm . 

No general method can be stated for finding the roots of (9), but 
fortunately it is possible to make approximations in practical problems 
involving metallic conductors which greatly simplify the solution. One 
will observe from (6) that if the conductivity is finite, a superposition of 
electric and magnetic type waves is necessary to satisfy the boundary con- 
Jitions, except in the symmetrical case n = 0. The field is not transverse, 
with respect to either the electric or the magnetic vector. If, however, 
the conductivity of either medium is infinite, the boundary conditions 
can be fulfilled by either a transverse magnetic or a transverse electric 
field. In this case the problem is completely determined by either the 
conditions 

(10) Si = 0, E, = 0, (r = a), 
or, alternatively, by 

(11) (rffi)-0 f gjff.-O, (r = a), 

taken from (14), page 484, noting that the tangential components of H 
do not vanish. In case the cylinder is a perfect conductor embedded in a 
dielectric, the allowed transverse magnetic and transverse electric modes 
are determined by roots respectively of the equations 



- = 

- U, u. 



The amplitudes oj and 6* are now quite independent and are determined 
solely by the nature of the excitation. In the inverse case of a dielectric 
cylinder bounded externally by a perfect conductor, the transverse 
magnetic and transverse electric modes are determined by the roots 
respectively of 

^(u) _ JM = 
J'M ~ 0> J.() u ' 



Sue. 9.16] CONDUCTOR EMBEDDED IN A DIELECTRIC 527 

In practice the conductivity of a metal, while not infinite, is yet so very 
large that the cylinder functions in such a region can be replaced by their 
asymptotic representations. It is then found that in each possible mode, 
either a transverse electric or transverse magnetic wave predominates. 
A small component of opposite type which spoils the transversality may 
be considered a correction term taking account of the finiteness of the 
conductivity. 

If only the symmetrical mode n = is excited, the field is independent 
of 6 and the coefficients a , 60 are independent of one another, whatever 
the conductivity. From (6) it is apparent that in this case the transverse 
magnetic wave satisfies 

Jo(ti) oj = H^(v) aj, 



, 

at the boundary, while for the transverse electric wave 
Vi(u) &$ = ?#?>(! 68, 

([K\ U V 

U ' Jo(u) b< = H$>(v) 6J. 

The allowed transverse magnetic modes are then determined by the 
roots A C m of the transcendental equation 

fcf J l (u) k\ 



while the transverse electric modes are found from 



u ti 

9.16. Conductor Embedded in a Dielectric. We shall discuss only 
the case of waves that are predominantly transverse magnetic, corre- 
sponding to the practically important problem of an axial current in a 
long, straight wire. 1 The axially symmetric solution, n = 0, shall be 
considered first. It will then be shown that the asymmetric modes, 
if excited, are immediately damped out and, consequently, never play 
a part in the propagation of current along a solid conductor. 

Suppose, first, that the conductivity of the cylinder is infinite. 
According to (12) the propagation factor is then determined by 

(18) * ( o - o 

u; - " 



1 In the case of transverse electric waves the lines of current are circles concentric 
With the axis of the conductor. This solution may, therefore, be applied to the study 
of propagation along an idealized solenoid. Cf. Sommerfeld, Ann. Physik, 15, 673, 
1904. If the conductor is solid the transverse electric modes are damped out. 



528 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

If v is very small, Hp(v) can be represented approximately by 



where 7 = 1.781. These expressions are obtained by evaluating the 
integral (39), page 367, in the neighborhood of the origin. The Hankel 
function is evidently multivalued, and as in the case of the logarithm we 
use only its principal value. This principal value can be defined most 
easily in terms of the asymptotic expression 

(20) H<(v) c 

which holds when v 1, p v. If one puts v = \v\e*+, then the 
principal value of H^ l) (v) lies in the domain ir/2 < <j> < 3?r/2. The 
principal value or branch of H^(v) vanishes at all infinite points of the 
positive-imaginary half plane. It is only with the roots of the principal 
branch that we are concerned, and it can be shown that the principal 
branches of Hp(v) and Hp(v) have none. 1 Consequently, the ratio 

Z7(l)f.A v,i 

. ~ _ i n g = o 



has only the solution v = 0, and hence h = & 2 . // the cylinder is infinitely 
conducting, the field is propagated in the axial direction with a velocity 
determined solely by the external medium. If this is free space, the waves 
travel along the cylinder without attenuation and with the velocity of 
light. 

In case the conductivity of the cylinder, while not infinite, is very 
large, as of a metal, h differs from fc 2 but there will be at least one mode 
for which the difference is small, so that v = a -\/k\ h 2 is a small quan- 
tity. The approximation (21) still applies to the right side of (16). 
On the left side we note that \ki\y> fc 2 ; since kt~*\h\, w~fcial. 
The Bessel functions J<*(u) and Ji(u) can, therefore, be replaced by their 
asymptotic representations. The imaginary part of u is positive. One 
will remember that ki = ai + ifti and that, if <n is very large, i c^ fa. 
Then in virtue of (18), page 359, 



(22) lim = lim tan a 

v ' ^-JoCw) *-* L 

Equation (16) now assumes the form 



whose right-hand member is independent of h. 
1 SOMMBRFBLD, in Riemaim- Weber, loc. dt. t p. 461. 



SEC. 9.16] CONDUCTOR EMBEDDED IN A DIELECTRIC 529 

This equation for h can be solved easily by successive approximations. 
The method proposed by Sommerfeld 1 goes as follows. Let 

- ty 2 ^ k * a 

'-"TSV 

In = ij. 

The quantity ?; is known and is to be determined. Now the logarithm 
of is a slowly varying function with respect to itself. Hence, if the 
nth approximation has been found, the (n + l)th is obtained from 

(25) n+i In n = 77, 
or 

(26) J = ^ 



In 2- 



In 2- 



In 2- 



ln-2- 



The convergence can be demonstrated for complex as well as real values 
of 77. Roots of (24) which lie near = 1 or which are associated with 
other branches than the principal of the logarithm are rejected. It turns 
out that In is a negative number of the order 20, and the convergence 
is hastened by assuming this value for In . From (24), with suitable 
approximations, 

(27) rj ~ (1 + {)1.75 X 10-"a? , 

where v is the frequency and a is expressed in meters. 

To illustrate, the following numerical example is quoted from Som- 
merfeld. 2 The wire is of copper, 1 mm. in radius. The frequency is 
10 9 cycles/sec, and the outside medium is air. Then (27) gives 

17 = -(1 + i)7.25 X 10~ 7 . 
Assuming In = 20, we obtain for the first approximation 

fc = (1 + 03.6 X 10- 8 , 
In fc = -16.8 + i| = -(16.8 - 0.78i). 

1 G^ttinger Nachr., 1899. Described in Riemann- Weber, loc. cit., p. 529. 
2 KIEMANN- WEBER, loc. cit., p. 530. 



530 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

Similarly 

fc = jJL- = (4.1 + 4.5010-', 

In {, = -(16.7 -0.83f), 

?s = j^ = (4.2 + 4.5i)10-8. 

3 differs but slightly from 2 and the series may be broken off here. 
The smallness of justifies the initial approximations for the Bessel and 
Hankel functions. Returning now to (24), we calculate the value of v, 
and hence of the propagation factor A, for a = 1 mm. 

h ~ & 2 [1 + (6.0 + 6.4i)10- 6 ] = a + ift. 
The phase velocity along the wire is 

F~c(l - 6.0 X 10- 5 ), 

while the distance the wave must travel to be reduced to 1/e of its initial 
amplitude is z = 770 meters. Although the velocity differs an inappreci- 
able amount from that of free space, the damping at this high frequency 
and for this size of wire (about No. 12, Brown and Sharpe gauge) is rela- 
tively high from the point of view of a communication line. 

If the radius a is extremely small, the quantity u^aki may also be 
small in spite of a large conductivity. To illustrate this point, Sommer- 
feld has worked out the case of a platinum wire, conductivity one-eighth 
that of copper, a = 2/x = 2 X 10~ 6 meter and v = 3 X 10 8 cycles/sec. 
One then finds that w ~ (1 + i)0.19, and consequently the Bessel 
functions JQ(U) and J\(u) cannot be replaced by their asymptotic expres- 
sions. In the case cited the value of u is so small that one can put 
approximately JQ(U) = 1, Ji(u) = u/2, and proceed according to the 
same method outlined above. One obtains 

h c- 0.083 + 0.061 i. 

The phase velocity proves to be only 76 per cent of its free-space value 
and the attenuation is so large that the amplitude is reduced to 1/e of its 
initial value in a distance z 16 cm. 

Thus far we have considered only one root of the determinantal 
equation (16), the root which in the case of high conductivity makes 
hc^k 2 and leads to what will be termed the principal wave. There are 



other roots, however, for which, v = a V&i ~ h 2 is very large. With 
the help of the asymptotic expressions for the Hankel functions, (22), 
page 359, we may now approximate (16) by 



If the left side of (28) is very large, the roots are given to the first approxi- 



SEC. 9.17] FURTHER DISCUSSION OF THE PRINCIPAL WAVE 531 

mation by Ji(u) = 0. But the roots of Ji(u) lie on the real axis of the 
w-plane. Consequently the imaginary part of h 2 must be equal to the 
imaginary part of k\. This means that in the case of metallic conductors 
the attenuation factor ft of h = a + i@ is of the order 2 X 10~ 3 v^/^, 
and hence the complementary waves corresponding to the other roots of (16) 
are damped out at once. 

The roots of (17), which give the modes of axially symmetric transverse 
electric waves can be investigated according to the same procedure. 
However, since there is no " magnetic conductivity," there is also no 
principal wave characterized by low attenuation. All axially symmetric 
transverse electric waves in a solid metal cylinder are damped out too 
quickly to be observable. 

In the asymmetric case we have seen that the field is a linear com- 
bination of transverse magnetic and transverse electric modes. The 
existence of a principal wave with small attenuation implies that h ~ & 2 , 
t><$C 1. Since n is now different from zero, this would lead to extremely 
large values of the right-hand member of (9), a condition that could be 
fulfilled only by nearly infinite values of the conductivity <n. It turns 
out that even with a conductivity as high as that of copper, the imaginary 
part of the propagation factor h is still very large and the attenuation 
such as to preclude the existence of asymmetric waves at any finite 
distance from the source. 

The results of this study may be summed up as follows. There are an 
infinite number of modes of propagation along a solid conducting cylinder. 
The amplitudes of these modes are the coefficients a n and b n , whose values 
are determined by the initial excitation the nature of the source. Of 
all these modes, however, only one, the principal mode, a symmetric, 
transverse magnetic wave, possesses a relatively low attenuation. All 
others are damped out within a centimeter or two from the source, even at 
frequencies as low as 60 cycles/sec. We shall see in the next section that 
the field of the principal wave within the conductor tends to concentrate 
in a thin skin near the surface. At high frequencies the center of the 
cylinder is essentially free of field and current, with a correspondingly 
small conversion of energy into heat. On the other hand the field and 
current distribution within the cylinder in the case of complementary 
waves is nearly uniform. The "skin effect" is then external, and the 
conversion into heat correspondingly large. As the conductivity of the 
cylinder is decreased, the attenuation of the principal wave increases, 
while that of the complementary waves is diminished until finally, in the 
limit of a perfect dielectric cylinder, we find the principal wave to have 
vanished and the complementary waves propagated without attenuation. 

9.17. Further Discussion of the Principal Wave. We have seen that 
in the case of a solid conducting cylinder only the principal wave plays an 



532 BOUNDARY-VALUE PROBLEMS [CHAP. IX 

important role. Our next task is to put the results of the preceding 
section in the form best adapted to numerical discussion. The first step 
is to express the coefficients aj an d a? in terms of the total conduction